Substance

Using Hillier and Lieberman's Streamlined Procedure To Solve Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems

 

Jsun Yui Wong

The computer program listed below seeks to solve the following 2-objective nonlinear mixed-integer preemptive goal programming problem from Ignizio [47, pp. 209-210] with the streamlined procedure of Hillier and Lieberman [42, pp. 289-291]:

Minimize { (X(4)) , ( X(6)) }

subject to

X(1) ^ 2 + 2 * X(2) + X(3) - X(4) =20

X(1) + X(1) * X(2) + X(5) - X(6) =10

X(1)=0, 1, 2, ...

X(2)=0, 1, 2, ...

All variables are nonnegative.

One notes line 1320, which is 1320 P = -10 ^ 20 * X(4) - X(6).

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 6

121 A(J44) = FIX(RND * 5)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 30000)

129 FOR KKQQ = 1 TO 6

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 6)

144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 REM IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP
221 FOR J44 = 1 TO 6

230 IF X(J44) < 0 THEN 1670

231 NEXT J44
232 X(1) = INT(X(1))
233 X(2) = INT(X(2))

1252 X(3) = 20 + X(4) - X(1) ^ 2 - 2 * X(2)

1254 X(5) = 10 + X(6) - X(1) - X(1) * X(2)

1267 FOR J44 = 1 TO 6

1269 IF X(J44) < 0 THEN 1670
1270 NEXT J44

1293 X(1) = INT(X(1))
1295 X(2) = INT(X(2))

1320 P = -10 ^ 20 * X(4) - X(6)

1418 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -4D+200 THEN 1999
1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [103]. The complete output of a single run through JJJJ=-31997 is shown below:

2       2       12       0       4
0       0       -32000

3 0 11 2.122757576172983D-282
7 5.334281726365734D-262 -7.457039392538717D-262
-31999

1 4 11.00000319772738 3.197727382885525D-06
5.520649189382835 .5206491893828358 -319772738288553.1
-31998

1       1       17       0       8
0       0       -31997
.
.
.

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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 3 seconds, not including the time for “Creating .EXE file.”

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan [104] are noteworthy.

Acknowledgement

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

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Hillier and Lieberman's Streamlined Procedure for Solving Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems

 

Jsun Yui Wong

The computer program listed below seeks to solve the immediately following 2-objective nonlinear integer goal programming problem:

Minimize P1* X(4)

Minmize P2* X(6)

where P1>>P2

subject to

X(1) ^ 2 + 4 * X(2) +X(3)-X(4)=4

X(1) + X(2) ^ 2 + X(5) - X(6)= 10

X(1), X(2)>=0 and are integers

X(3) * X(4) = 0

X(5) * X(6) =0

X(3) through X(6)>=0.

The problem above is based on a problem in Malhotra and Arora [62, p. 48, Auxiliary Problem (P2)].

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 6

121 A(J44) = FIX(RND * 3)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 90000)

129 FOR KKQQ = 1 TO 6

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 6)

144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 REM IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP
201 FOR J44 = 1 TO 6

205 IF X(J44) < 0 THEN 1670

206 NEXT J44

214 X(1) = INT(X(1))

215 X(2) = INT(X(2))
221 IF X(3) * X(4) > 0## THEN 1670
224 IF X(5) * X(6) > 0## THEN 1670

236 GOTO 1244

1244 X(3) = 4 + X(4) - X(1) ^ 2 - 4 * X(2)

1250 X(5) = 10 + X(6) - X(1) - X(2) ^ 2

1267 FOR J44 = 1 TO 6
1268 REM X(J44) = INT(X(J44))

1269 IF X(J44) < 0 THEN 1670

1270 NEXT J44

1275 X(1) = INT(X(1))
1277 X(2) = INT(X(2))

1296 IF X(3) * X(4) > 0## THEN 1670
1298 IF X(5) * X(6) > 0## THEN 1670

1305 P = -10 ^ 40 * X(4) - X(6)

1418 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 6

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [103]. The complete output of a single run through JJJJ=-31996 is shown below:
0       1       0       0       9
0       0       -32000
2 2 2 1 0
2 -4D+250 -31999
1 2 0 5 5
0 -5D+40 -31998
1 0 3 0 9
0 0 -31997
0 1 0 0 9
0 0 -31996
.
.
.

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 [103], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Malhotra and Arora [62, p. 50].

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [104].

Acknowledgement

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

References

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Solving a Nonlinear Preemptive (Lexigraphic) Goal Programming Problem in Continuous Variables, Second Edition

 

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear goal programming problem in Baykasoglu [11, page 5-10 of Chapter 5, Example 1]:

Lexmin

{ Z1 = X(4) + X(6) + X(7) + X(8) + X(9) + X(10) + X(11) }

subject to

5.28 * X(1) ^ 2 + 3.74 * X(2) ^ 2 + 5.28 * X(3) ^ 2 + X(4) - X(5) =3.575

(.178 / X(1) ^ 2) + X(6) - X(7) =1.0

(.255 / X(2) ^ 2) + X(8) - X(9) =1.0

(.178 / X(3) ^ 2) + X(10) - X(11) =1.0

All variables are continuous and nonnegative.

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 11

121 A(J44) = .1 + RND

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 100000)

129 FOR KKQQ = 1 TO 11

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 11)

144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 REM IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP
221 FOR J44 = 1 TO 11

232 IF X(J44) < 0## THEN 1670

234 NEXT J44

1144 REM X(4) = 3.575 + X(5) - 5.28 * X(1) ^ 2 - 3.74 * X(2) ^ 2 - 5.28 * X(3) ^ 2
1145 X(6) = 1 + X(7) - (.178 / X(1) ^ 2)
1146 X(8) = 1 + X(9) - (.255 / X(2) ^ 2)

1149 X(10) = 1 + X(11) - (.178 / X(3) ^ 2)
1153 X(4) = 3.575 + X(5) - 5.28 * X(1) ^ 2 - 3.74 * X(2) ^ 2 - 5.28 * X(3) ^ 2
1155 FOR J44 = 1 TO 11
1158 IF X(J44) < 0## THEN 1670

1159 NEXT J44

1414 P = -X(4) - X(6) - X(7) - X(8) - X(9) - X(10) - X(11)

1417 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -.4419 THEN 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5)

1933 PRINT A(6), A(7), A(8), A(9), A(10), A(11), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-31369 is shown below:
.4219004637354153          .6732712030730998          .4219004621955346
1.093166158224086D-08          4.39770633389389D-17
7.30426132587985D-09          2.5286530975886D-19          .4374513307140895
7.515927122128481D-19          4.526660838700958D-12          4.12507632874798D-20
-.437451348954539          -31908
.4219004621945798 .673271206607752 .42190046219458
1.112011958209758D-15 3.610839300990171D-21
5.138576196495948D-16 2.184629418770705D-19 .437451336620814
5.011913593947958D-20 1.303427851762073D-15 3.180283587021886D-19
-.437451335520817 -31586
.4219004621947022 .67327120660747 .4219004621945837
8.191065013762877D-13 1.067958847455112D-18
5.807553873568572D-13 2.65657721500138D-19 .4374513366203561
4.525657679895911D-20 1.893466937061272D-14 6.903536636520767D-19
-.437451335521775 -31369
.
.
.

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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31369 was 2 minutes, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Baykasoglu [11, Table 5.1 on page 5-11 of Chapter 5].

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].

Acknowledgement

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

References

[0] Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013). Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013, Vol. 83, No. 5, pp. 962-976.

[1] Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2011). Integer goal programming approach for finding a compromise allocation of repairable components, International Journal of Engineering, Science and Technology 2011, Vol. 3, No. 6, pp. 184-195. https://www.researchgate.net> publication > 260348481

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Solving Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems Using Hillier and Lieberman's Streamlined Procedure

 

Jsun Yui Wong

1. A Problem from Markland and Sweigart [63]

The first computer program listed below seeks to solve the immediately following problem from Markland and Sweigart [63, pp, 338-339] with the streamlined procedure of Hillier and Lieberman [42, pp. 289-291]:

Minimize

P1* (X(4)) +P2 * X(6) +P3 * (X(9)) +P4* X(10)

where P1>>P2, P2>>P3, P3>>P4

subject to

X(1)+ X(4) - X(5) =6

X(2 + X(6) - X(7) =3

X(3) - X(1) - X(2) + X(8) - X(9) =0

12.5 * X(3) - 10 * X(2)+ X(10) - X(11) =30

X(1) + X(2) + X(3)=12

X(3) - .8 * X(2) >= 0

All variables are nonnegative.

The formulation above is due to Markland and Sweigart [63, pp. 803-804].

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 11

121 A(J44) = FIX(RND * 4)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 90000)

129 FOR KKQQ = 1 TO 11

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 11)

144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 REM IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP
221 FOR J44 = 1 TO 11

231 REM X(J44) = INT(X(J44))

232 IF X(J44) < 0 THEN 1670

234 NEXT J44
244 X(1) = 12 - X(2) - X(3)

250 X(4) = 6 + X(5) - X(1)
252 X(6) = 3 + X(7) - X(2)

253 X(8) = 0 + X(9) - X(3) + X(1) + X(2)

255 X(10) = 30 + X(11) - 12.5 * X(3) + 10 * X(2)
266 IF X(3) - .8 * X(2) < 0 THEN 1670

267 FOR J44 = 1 TO 11
268 REM X(J44) = INT(X(J44))

269 IF X(J44) < 0 THEN 1670
270 NEXT J44
1415 P = -10 ^ 40 * (X(4)) - 10 ^ 30 * X(6) - 10 ^ 20 * (X(9)) - X(10)

1417 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -2 * 10 ^ 10 THEN 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-31973 is shown below:
.
.
.
6       3       3       0       0
0       0       6       4.084060676929564D-36
22.5       1.448217364822858D-15       -22.5       -31989
6       3       3       0       0
0       0       6       0       22.5
8.234538890298874D-15       -22.5       -31973
.
.
.

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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31973 was 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Markland and Sweigart [63, pp. 803-804].

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].

2. A Mixed-Integer Version of the Same Problem

One notes the difference between line 144 above and line 144 below, which are 144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162 and 144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162, respectively.

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 11

121 A(J44) = FIX(RND * 4)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 90000)

129 FOR KKQQ = 1 TO 11

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 11)

144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 REM IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP
221 FOR J44 = 1 TO 11

231 REM X(J44) = INT(X(J44))

232 IF X(J44) < 0 THEN 1670

234 NEXT J44
244 X(1) = 12 - X(2) - X(3)

250 X(4) = 6 + X(5) - X(1)
252 X(6) = 3 + X(7) - X(2)

253 X(8) = 0 + X(9) - X(3) + X(1) + X(2)

255 X(10) = 30 + X(11) - 12.5 * X(3) + 10 * X(2)
266 IF X(3) - .8 * X(2) < 0 THEN 1670

267 FOR J44 = 1 TO 11
268 REM X(J44) = INT(X(J44))

269 IF X(J44) < 0 THEN 1670
270 NEXT J44
1415 P = -10 ^ 40 * (X(4)) - 10 ^ 30 * X(6) - 10 ^ 20 * (X(9)) - X(10)

1417 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 11

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -2 * 10 ^ 10 THEN 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-31995 is shown below:
6       3       3       0       0
0       0      6       0       22.5
0       -22.5       -32000
6 3 3 0 0
0 0 6 0 22.5
0 -22.5 -31999
6 3 3 0 0
0 0 6 0 22.5
0 -22.5 -31997
6 3 3 0 0
0 0 6 0 22.5
0 -22.5 -31996
6 3 3 0 0
0 0 6 0 22.5
0 -22.5 -31995
.
.
.

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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Markland and Sweigart [63, pp. 803-804].

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].

Acknowledgement

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

References

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Hillier and Liberman's Streamlined Procedure for Solving Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems

 

Jsun Yui Wong

The computer program listed below seeks to solve the immediately following problem with the streamlined procedure of Hillier and Lieberman [42, pp. 289-291]:

Minimize

X(3)

minimize

1*X(5)

minimize

2*X(6)

minimize

X(8)

these Xs are deviational variables. (Please see line 1416, which is 1416 P = -10 ^ 10 * X(3) - 1 * 10 ^ 5 * (X(5)) - 2 * 10 ^ 5 * X(6) - X(8). )

subject to

X(3) - X(4) + 4 * X(1) + 3 * X(2)=500
X(5) + X(1)=50

X(6) + X(2)=75

X(7) - X(8) + X(1) + 2 * X(2)-120

All variales are nonnegative and are integers.

The problem above is the problem on p. 47 of Taylor, McLaren, and McLaren [92, p. 47, Problem 2] plus the restriction that all variables are integer variables.

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 8

121 A(J44) = FIX(RND * 5)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 100000)

129 FOR KKQQ = 1 TO 8

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 8)

144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 REM GOTO 162
154 REM IF j > 4.5 THEN GOTO 156 ELSE GOTO 162

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

169 NEXT IPP

221 FOR J44 = 1 TO 8

231 X(J44) = INT(X(J44))

232 IF X(J44) < 0 THEN 1670

234 NEXT J44

260 X(3) = 500 + X(4) - 4 * X(1) - 3 * X(2)
262 X(5) = 50 - X(1)

264 X(6) = 75 - X(2)

266 X(7) = 120 + X(8) - X(1) - 2 * X(2)
267 FOR J44 = 1 TO 8

268 X(J44) = INT(X(J44))

269 IF X(J44) < 0 THEN 1670
270 NEXT J44

1416 P = -10 ^ 10 * X(3) - 1 * 10 ^ 5 * (X(5)) - 2 * 10 ^ 5 * X(6) - X(8)

1417 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 REM

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [100]. The complete output of a single run through JJJJ=-31994 is shown below:

50       75       75       0       0
0       0       80       -750000000080
-32000

50 75 75 0 0
0 0 80 -750000000080
-31999

50 75 75 0 0
0 0 80 -750000000080
-31998

50 75 75 0 0
0 0 80 -750000000080
-31997

50 75 75 0 0
0 0 80 -750000000080
-31996

50 75 75 0 0
0 0 80 -750000000080
-31995

50 36 192 0 0
39 0 2 -1920007800002
-31994

.
.
.

 

One notes that at JJJJ=-32000, M=-750000000080.

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 [100], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Taylor, McLaren, and McLaren [92, p. 49, Table 9.4].

Remark 4 on page 198 of Winston and Venkataramanan is noteworthy [101].

Acknowledgment

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

References

[1] Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013). Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013, Vol. 83, No. 5, pp. 962-976.

[2] Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2011). Integer goal programming approach for finding a compromise allocation of repairable components, International Journal of Engineering, Science and Technology 2011, Vol. 3, No. 6, pp. 184-195. https://www.researchgate.net> publication > 260348481

[3] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[6] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[7] Oscar Augusto, Bennis Fouad, Stephane Caro (2012). A new method for decision making in multi-objective optimization problems. Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[8] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

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[10] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

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Solving an Integer Nonlinear Programming Problem

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from page 192 of Ali, Raghav, and Bari [1, p. 192 for i=4]:

For i=4

Maximize

(1 - (1-r(4)) ^ (n(4)-a(4) + X(4)))

where r(4), n(4), and a(4) are given--please see Table 1 of Ali, Raghav, and Bari [1, p. 192 for i=4]

subject to

(3* (X(1) + EXP(.25 * X(1)))) + (4 * (X(2) + EXP(.25 * X(2)))) + (3 * (X(3) + EXP(.25 * X(3)))) + (5 * (X(4) + EXP(.25 * X(4)))) + (4 * (X(5) + EXP(.25 * X(5)))) <= 60

(8 * (X(1) + EXP(.25 * X(1)))) + (7 * (X(2) + EXP(.25 * X(2)))) + (8 * (X(3) + EXP(.25 * X(3)))) + (4 * (X(4) + EXP(.25 * X(4)))) + (6 * (X(5) + EXP(.25 * X(5)))) <= 90

0<= X(i) <=a(i), i=1, 2, 3, 4, 5 and integers.

0 DEFDBL A-Z
1 REM DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 5

122 A(J44) = FIX(RND * 3)

123 NEXT J44

128 FOR I = 0 TO FIX(RND * 1000)

129 FOR KKQQ = 1 TO 5

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

143 j = 1 + FIX(RND * 5)
144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162

145 GOTO 162
154 IF j > 5.5 THEN GOTO 156 ELSE GOTO 162

156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * r

161 GOTO 169

162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 2.3) ELSE X(j) = A(j) + FIX(1 + RND * 2.3)

169 NEXT IPP

212 IF X(1) > 3 THEN 1670

213 IF X(2) > 2 THEN 1670

214 IF X(3) > 2 THEN 1670

216 IF X(4) > 2 THEN 1670

217 IF X(5) > 3 THEN 1670
231 FOR J44 = 1 TO 5

232 X(J44) = INT(X(J44))

233 IF X(J44) < 0 THEN 1670
255 NEXT J44

304 IF (3 * (X(1) + EXP(.25 * X(1)))) + (4 * (X(2) + EXP(.25 * X(2)))) + (3 * (X(3) + EXP(.25 * X(3)))) + (5 * (X(4) + EXP(.25 * X(4)))) + (4 * (X(5) + EXP(.25 * X(5)))) > 60 THEN 1670

306 IF (8 * (X(1) + EXP(.25 * X(1)))) + (7 * (X(2) + EXP(.25 * X(2)))) + (8 * (X(3) + EXP(.25 * X(3)))) + (4 * (X(4) + EXP(.25 * X(4)))) + (6 * (X(5) + EXP(.25 * X(5)))) > 90 THEN 1670

1307 P = (1 - (.2) ^ (2 + X(4)))

1311 IF P <= M THEN 1670
1420 M = P

1442 FOR KLX = 1 TO 5

1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -999999999 GOTO 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5)

1939 PRINT M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [97]. The output of a single run through JJJJ= -31961 is summarized below:
0 2 1 2 1
.9984 -32000
0 1 2 2 1
.9984 -31999
2 0 1 2 1
.9984 -31998
2 1 0 2 2
.9984 -31997
.
.
.
1 1 1 2 1
.9984 -31961
.
.
.

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 [97], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31961 was 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Ali, Raghav, and Bari [1, p. 192].

The other cases of i=1, 2, 3, and 5 can be done similarly.

Acknowledgment

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

References

[0] Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013). Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013, Vol. 83, No. 5, pp. 962-976.

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