Jsun Yui Wong

The computer program listed below seeks to solve the following 4-objective integer program from Hwang, Lee, Tillman, and Lie [38, p. 436, Example 2] and Lee [45, p. 77 and p. 84, Example 3-4]:

Minimize {(X(9)), (X(11)), (X(12)), (X(10))}, which have priority 1 through priority four, respectively

X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2 + X(6)-X(10)=110

7 * (X(1) + EXP(X(1) / 4)) + 7 * (X(2) + EXP(X(2) / 4)) + 5 * (X(3) + EXP(X(3) / 4)) + 9 * (X(4) + EXP(X(4) / 4)) + 4 * (X(5) + EXP(X(5) / 4)) + X(7)-X(11)=175

7 * (X(1) * EXP(X(1) / 4)) + 8 * (X(2) * EXP(X(2) / 4)) + 8 * (X(3) * EXP(X(3) / 4)) + 6 * (X(4) * EXP(X(4) / 4)) + 9 * (X(5) * EXP(X(5) / 4)) + X(8)-X(12)=200

(1 - (1 - .80) ^ X(1)) * (1 - (1 - .85) ^ X(2)) * (1 - (1 - .90) ^ X(3)) * (1 - (1 - .65) ^ X(4)) * (1 - (1 - .75) ^ X(5)) + X(9)-X(13)=.99

where X(1) through X(5) are positive general integer variables and X(6) through X(13) are >=0.

Whereas line 1899 of the earlier edition is 1899 IF AAA999 < .001 THEN 1936 ELSE GOTO 1999, here line 1899 is 1899 IF AAA999 < .0000000001 THEN 1936 ELSE GOTO 1999.

0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

86 M = -3E+50

92 A(1) = FIX(RND * 9)

93 A(2) = FIX(RND * 9)

94 A(3) = FIX(RND * 9)

96 A(4) = FIX(RND * 9)

97 A(5) = FIX(RND * 9)

111 FOR J44 = 6 TO 13

114 A(J44) = (RND * 50)

117 NEXT J44

118 A(9) = (RND * .5)

119 A(13) = (RND * .5)

128 FOR I = 1 TO 3000

129 FOR KKQQ = 1 TO 13

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 5)

153 J = 1 + FIX(RND * 13)

154 IF J > 5 THEN GOTO 156 ELSE GOTO 162

155 REM 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) - INT(RND * 4) ELSE X(J) = A(J) + INT(RND * 4)

164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

169 NEXT IPP

172 X(1) = INT(X(1))

174 X(2) = INT(X(2))

176 X(3) = INT(X(3))

178 X(4) = INT(X(4))

179 X(5) = INT(X(5))

188 IF X(1) < 1 THEN 1670

189 IF X(2) < 1 THEN 1670

190 IF X(3) < 1 THEN 1670

192 IF X(4) < 1 THEN 1670

194 IF X(5) < 1 THEN 1670

208 IF X(1) > 9 THEN 1670

209 IF X(2) > 9 THEN 1670

210 IF X(3) > 9 THEN 1670

211 IF X(4) > 9 THEN 1670

212 IF X(5) > 9 THEN 1670

215 FOR J44 = 6 TO 13

216 IF X(J44) < 0 THEN 1670

219 NEXT J44

311 X(10) = -110 + X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2 + X(6)

315 X(11) = -175 + 7 * (X(1) + EXP(X(1) / 4)) + 7 * (X(2) + EXP(X(2) / 4)) + 5 * (X(3) + EXP(X(3) / 4)) + 9 * (X(4) + EXP(X(4) / 4)) + 4 * (X(5) + EXP(X(5) / 4)) + X(7)

319 X(12) = -200 + 7 * (X(1) * EXP(X(1) / 4)) + 8 * (X(2) * EXP(X(2) / 4)) + 8 * (X(3) * EXP(X(3) / 4)) + 6 * (X(4) * EXP(X(4) / 4)) + 9 * (X(5) * EXP(X(5) / 4)) + X(8)

322 X(13) = -.99 + (1 - (1 - .80) ^ X(1)) * (1 - (1 - .85) ^ X(2)) * (1 - (1 - .90) ^ X(3)) * (1 - (1 - .65) ^ X(4)) * (1 - (1 - .75) ^ X(5)) + X(9)

326 FOR J44 = 10 TO 13

328 IF X(J44) < 0 THEN 1670

330 NEXT J44

437 PDU = -200000000 * X(9) - 10000000 * X(11) - 100000 * X(12) - X(10) - 9999999999 * X(6) * X(10) - 9999999999 * X(7) * X(11) - 9999999999 * X(8) * X(12) - 9999999999 * X(9) * X(13)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 13

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1555 AAA999 = X(9)

1557 GOTO 128

1670 NEXT I

1889 REM IF M < -109900 THEN 1999

1899 IF AAA999 < .0000000001 THEN 1936 ELSE GOTO 1999

1936 PRINT A(6), A(7), A(8), A(9)

1946 PRINT A(10), A(11), A(12), A(13)

1956 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1966 PRINT AAA999

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [78]. The complete output of a single run through JJJJ= -31567 is shown below:

5.084848658856128D-20 7.274376618665705D-20 1.411973390976466D-20

1.53119137540907D-16

115 55.01540412849394 275.680631915885

6.923585072159761D-04

4 4 3 5 5

-577722219.476528 -31861

1.53119137540907D-16

2.90886936074791D-20 4.594354107852364D-20 1.436075374704454D-20

1.959098359292364D-16

115 55.01540412849394 275.680631915885

6.92358507216019D-04

4 4 3 5 5

-577722219.476528 -31567

1.959098359292364D-16

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 [78], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31567 was 94 seconds, not including the time for “Creating .EXE file" (100 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Hwang, Lee, Tillman, and Lie [38, p. 436, Example 2].

**Acknowledgment**

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

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