Generating Feasible Solutions of a Three-Objective Integer Linear Program

Jsun Yui Wong

The computer program listed below seeks to generate all or some of the feasible solutions of the following integer program with three objectives:

Maximize      (3 * X(1) + X(2) + 2 * X(3) + X(4))

maximize      (X(1) – X(2) + 2 * X(3) + 4 * X(4))

maximize      (-X(1) + 5 * X(2) + X(3) + 2 * X(4))

subject to

2 * X(1) + X(2) + 4 * X(3) + 3 * X(4) <= 60

3 * X(1) + 4 * X(2) + X(3) + 2 * X(4) <= 60

X(1) through X(4)>=0 and are general integer variables.

The problem above is based on p. 661 of Zionts and Wallenius [86].
0 REM 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

87 M = -4E+250

98 W1 = RND

99 W2 = RND

101 IF W1 + W2 > 1## THEN 1670

103 W3 = 1 – W1 – W2

111 FOR J44 = 1 TO 4
113 A(J44) = FIX(RND * 11)

115 REM A(J44) = RND

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 4001)

129 FOR KKQQ = 1 TO 4

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

135 FOR IPP = 1 TO FIX(1 + RND * 3.3)
143 J = 1 + FIX(RND * 4)
154 REM IF RND < .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) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

293 FOR J44 = 1 TO 4
294 IF X(J44) < 0## THEN 1670

296 X(J44) = INT(X(J44))

297 NEXT J44
324 IF 2 * X(1) + X(2) + 4 * X(3) + 3 * X(4) > 60## THEN 1670

334 IF 3 * X(1) + 4 * X(2) + X(3) + 2 * X(4) > 60## THEN 1670

539 P = W1 * (3 * X(1) + X(2) + 2 * X(3) + X(4)) + W2 * (X(1) – X(2) + 2 * X(3) + 4 * X(4)) + W3 * (-X(1) + 5 * X(2) + X(3) + 2 * X(4))
1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1457 OBJ1 = (3 * X(1) + X(2) + 2 * X(3) + X(4))

1458 OBJ2 = (X(1) – X(2) + 2 * X(3) + 4 * X(4))
1459 OBJ3 = (-X(1) + 5 * X(2) + X(3) + 2 * X(4))

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 REM PRINT W1, W2, W3

1934 PRINT A(1), A(2), A(3), A(4)

1936 PRINT OBJ1, OBJ2, OBJ3, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [84]. The output of a single run through JJJJ= -31675 is summarized below:

14 0 2 8
54 50 4 -32000

3 1 13 0
36 28 15 -31999

6 0 0 16
34 70 26 -31997

12 0 0 12
48 60 12 -31996

18 0 6 0
66 30 -12 -31994

0 10 8 6
32 30 70 -319 87

0 10 8 6
32 30 70 -319 86

0 12 12 0
36 12 72 -31984

0 12 12 0
36 12 72 -31983

12 0 0 12
48 60 12 -31981

12 0 0 12
48 60 12 -31978

0 9 6 9
30 39 69 -31977

0 8 4 12
28 48 68 -31976

0 9 6 9
30 39 69 -31975

6 10 0 1
29 0 46 -31973

3 6 3 12
33 51 54 -31971

4 0 0 17
29 72 30 -31969

7 5 0 8
34 34 34 -31968

0 1 8 9
26 51 31 -31966

0 7 2 15
26 57 67 -31964

5 5 3 11
37 50 45 -31959

10 3 4 7
48 43 23 -31955

0 8 4 12
28 48 68 -31954

0 10 8 6
32 30 70 -31953

0 6 0 18
24 66 66 -31952

0 12 12 0
36 12 72 -31951

2 5 0 17
28 65 57 -31949

9 0 0 14
41 65 19 -31948

0 6 0 18
24 66 66 -31947

0 12 12 0
36 12 72 -31946

6 8 10 0
46 18 44 -31945

0 0 15 0
30 30 15 -31943

.
.
.

0 15 0 0
15 -15 75 -31922

.
.
.
20 0 0 0
60 20 -20 -31675

.
.
.
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 [84], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31675 was 3 seconds, not including the time for “Creating .EXE file” (18 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Zionts and Wallenius [86, p. 661, Table 1].

Acknowledgment

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

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Generating Feasible Solutions of a Four-Objective Mixed-Integer Nonlinear Program

Jsun Yui Wong

The computer program listed below seeks to generate all or some of the feasible solutions of the following rocket injector design problem with four objectives:

Minimize [F1, F2, F3, F4]

where

F1 = .692 + .477 * X(1) – .687 * X(4) – .08 * X(3) – .0652 * X(2) – .167 * X(1) ^ 2 – .0129 * X(1) * X(4) + .0796 * X(4) ^ 2 – .0634 * X(1) * X(3) – .0257 * X(3) * X(4) + .0877 * X(3) ^ 2 – .0521 * X(1) * X(2) + .00156 * X(2) * X(4) + .00198 * X(2) * X(3) + .0184 * X(2) ^ 2
F2 = .37 – .205 * X(1) + .0307 * X(4) + .108 * X(3) + 1.019 * X(2) – .135 * X(1) ^ 2 + .0141 * X(1) * X(4) + .0998 * X(4) ^ 2 + .208 * X(1) * X(3) – .0301 * X(3) * X(4) – .226 * X(3) ^ 2 + .353 * X(1) * X(2) – .0497 * X(2) * X(3) – .423 * X(2) ^ 2 + .202 * X(1) ^ 2 * X(4) – .281 * X(1) ^ 2 * X(3) – .342 * X(1) * X(4) ^ 2 – .245 * X(3) * X(4) ^ 2 + .281 * X(3) ^ 2 * X(4) – .184 * X(1) * X(2) ^ 2 + .281 * X(1) * X(3) * X(4)

F3 = .153 – .322 * X(1) + .396 * X(4) + .424 * X(3) + .0226 * X(2) + .175 * X(1) ^ 2 + .0185 * X(1) * X(4) – .0701 * X(4) ^ 2 – .251 * X(1) * X(3) + .179 * X(3) * X(4) + .015 * X(3) ^ 2 + .0134 * X(1) * X(2) + .0296 * X(2) * X(4) + .0752 * X(2) * X(3) + .0192 * X(2) ^ 2

F4 = .758 + .358 * X(1) – .807 * X(4) + .0925 * X(3) – .0468 * X(2) – .172 * X(1) ^ 2 + .0106 * X(1) * X(4) + .0697 * X(4) ^ 2 – .146 * X(1) * X(3) – .0416 * X(3) * X(4) + .102 * X(3) ^ 2 – .0694 * X(1) * X(2) – .00503 * X(2) * X(4) + .0151 * X(2) * X(3) + .0173 * X(2) ^ 2

subject to

X(1)=.01* X(5), 0<= X(5) <=100, and X(5) an integer
X(2)=.01* X(6), 0<= X(6) <=100, and X(6) an integer
X(3)=.01* X(7), 0<= X(7) <=100, and X(7) an integer
X(4)=.01* X(8), 0<= X(8) <=100, and X(8) an integer

0<= X(1) through X(4)<=1.

The formulation above is based on Burachik, Kaya, and Rizvi [11, pp. 13-14], especially on their x(1)=0.2* x(1) similar, 0<= x(1) similar <=3, and x(1) similar an integer.
0 REM 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

87 M = -4E+250

89 W1 = FIX(RND * 11) * .1##
91 W2 = FIX(RND * 11) * .1##

95 IF (W1 + W2) > 1 THEN 1999
96 W3 = FIX(RND * 11) * .1##

97 IF (W1 + W2 + W3) > 1 THEN 1999
99 W4 = 1 – W1 – W2 – W3

117 FOR J44 = 5 TO 8
119 A(J44) = FIX(RND * 101)
121 NEXT J44
128 FOR I = 0 TO FIX(RND * 21)
129 FOR KKQQ = 5 TO 8

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

135 FOR IPP = 1 TO FIX(1 + RND * .3)
143 J = 5 + FIX(RND * 4)
155 GOTO 166
156 R = (1 – RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * R

161 GOTO 169

162 REM IF RND < .5 THEN X(j) = A(j) – FIX(1 + RND * .3) ELSE X(j) = A(j) + FIX(1 + RND * .3)

166 REM

167 IF RND < .5 THEN X(J) = A(J) – FIX(1 + RND * 1.3) ELSE X(J) = A(J) + FIX(1 + RND * 1.3)
168 X(J – 4) = .01 * X(J)
169 NEXT IPP
173 FOR J44 = 5 TO 8

174 X(J44) = INT(X(J44))
177 NEXT J44
293 FOR J44 = 5 TO 8

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

296 IF X(J44) > 100## THEN 1670
297 NEXT J44

298 FOR J44 = 1 TO 4
299 IF X(J44) < 0## THEN 1670

301 IF X(J44) > 1## THEN 1670

304 NEXT J44
401 F1 = .692 + .477 * X(1) – .687 * X(4) – .08 * X(3) – .0652 * X(2) – .167 * X(1) ^ 2 – .0129 * X(1) * X(4) + .0796 * X(4) ^ 2 – .0634 * X(1) * X(3) – .0257 * X(3) * X(4) + .0877 * X(3) ^ 2 – .0521 * X(1) * X(2) + .00156 * X(2) * X(4) + .00198 * X(2) * X(3) + .0184 * X(2) ^ 2
412 F2 = .37 – .205 * X(1) + .0307 * X(4) + .108 * X(3) + 1.019 * X(2) – .135 * X(1) ^ 2 + .0141 * X(1) * X(4) + .0998 * X(4) ^ 2 + .208 * X(1) * X(3) – .0301 * X(3) * X(4) – .226 * X(3) ^ 2 + .353 * X(1) * X(2) – .0497 * X(2) * X(3) – .423 * X(2) ^ 2 + .202 * X(1) ^ 2 * X(4) – .281 * X(1) ^ 2 * X(3) – .342 * X(1) * X(4) ^ 2 – .245 * X(3) * X(4) ^ 2 + .281 * X(3) ^ 2 * X(4) – .184 * X(1) * X(2) ^ 2 + .281 * X(1) * X(3) * X(4)
415 F3 = .153 – .322 * X(1) + .396 * X(4) + .424 * X(3) + .0226 * X(2) + .175 * X(1) ^ 2 + .0185 * X(1) * X(4) – .0701 * X(4) ^ 2 – .251 * X(1) * X(3) + .179 * X(3) * X(4) + .015 * X(3) ^ 2 + .0134 * X(1) * X(2) + .0296 * X(2) * X(4) + .0752 * X(2) * X(3) + .0192 * X(2) ^ 2
419 F4 = .758 + .358 * X(1) – .807 * X(4) + .0925 * X(3) – .0468 * X(2) – .172 * X(1) ^ 2 + .0106 * X(1) * X(4) + .0697 * X(4) ^ 2 – .146 * X(1) * X(3) – .0416 * X(3) * X(4) + .102 * X(3) ^ 2 – .0694 * X(1) * X(2) – .00503 * X(2) * X(4) + .0151 * X(2) * X(3) + .0173 * X(2) ^ 2
530 P = W1 * (-F1) + W2 * (-F2) + W3 * (-F3) + W4 * (-F4)
1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 8

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1471 OBJ1 = F1

1472 OBJ2 = F2

1473 OBJ3 = F3

1474 OBJ4 = F4

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1929 PRINT A(1), A(2), A(3), A(4)

1933 PRINT OBJ1, OBJ2, OBJ3, OBJ4, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [84]. The feasible solutions of a single run through JJJJ= 32000–the end of the computer program above–are summarized below:

0 0 .72 0
.6798637 3306016 466056 .8774768 -31999

.18 .71 .71 .94
.1153403 .889521 .8952287 .1632167 -31996

.
.
.
.57       .21       .13       .94
.2917632       .460916       .4202703       .1947573       31965

.57 .2 .49 .94
.2618301 .4706542 .5899841 .2086946 31976

.87 .57 .47 .58
.4952626 .7313005 .4252029 .4418888 31980

.87 .34 .53 .52
.5501343 .554319 .3958332 .5099578 31988

.55 .08 .37 .98
.25530429 .3370665 .5393924 .814226 31989

.55 .61 .37 .97
.2167015 .7802386 .5888813 .1499008 31991

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 [84], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = 32000 was 3 seconds, not including the time for “Creating .EXE file” (18 seconds, total, including the time for “Creating .EXE file”).

Acknowledgment

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

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Solving a Multiple-Objective Integer Nonlinear Program, Second Edition

Jsun Yui Wong

The present problem is based on the following multiple-objective nonlinear programming problem in Siwale [66, p. 3, Example 1]:

Minimize 1 – EXP(-SUM1), where SUM1 is defined by line 221 below

minimize 1 – EXP(-SUM2), where SUM2 is defined by line 225 below

subject to

-2 <=X(i) <=2, i=1, 2, 3,…, 8.

The above problem of 8 continuous variables has been modified to the following problem with 8 general integer variables by the following line 115, which is

115 A(J44) = -2 + INT(RND * 4).

One notes the contrast between line 422 of the earlier edition (February 20, 2019), which is 422 PDU =-1 + EXP(-SUM1), and line 431 here, which is
431 PDU = W1 * (-1 + EXP(-SUM1)) + W2 * (-1 + EXP(-SUM2)).

 

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

87 M = -3E+50

92 EPSILON2 = RND
93 IF RND < 1 / 11 THEN W1 = 0 ELSE IF RND < 1 / 10 THEN W1 = .1 ELSE IF RND < 1 / 9 THEN W1 = .2 ELSE IF RND < 1 / 8 THEN W1 = .3 ELSE IF RND < 1 / 7 THEN W1 = .4 ELSE IF RND < 1 / 6 THEN W1 = .5 ELSE IF RND < 1 / 5 THEN W1 = .6 ELSE IF RND < 1 / 4 THEN W1 = .7 ELSE IF RND < 1 / 3 THEN W1 = .8 ELSE IF RND < 1 / 2 THEN W1 = .9 ELSE W1 = 1

99 W2 = 1 – W1

111 FOR J44 = 1 TO 8

115 A(J44) = -2 + INT(RND * 4)
121 NEXT J44

128 FOR I = 1 TO FIX(1 + RND * 100)
129 FOR KKQQ = 1 TO 8

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

135 FOR IPP = 1 TO FIX(1 + RND * 3)
143 j = 1 + FIX(RND * 8)
154 REM GOTO 162
156 R = (1 – RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * R

161 GOTO 169

162 REM IF RND < .5 THEN X(j) = A(j) – FIX(1 + RND * .3) ELSE X(j) = A(j) + FIX(1 + RND * .3)
164 IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

173 FOR J44 = 1 TO 8

174 X(J44) = INT(X(J44))

175 NEXT J44
176 FOR J44 = 1 TO 8

177 IF X(J44) < -2## THEN 1670

187 IF X(J44) > 2## THEN 1670
191 NEXT J44

217 SUM1 = 0
219 FOR J44 = 1 TO 8
221 SUM1 = SUM1 + (X(J44) – 1 / 8 ^ .5##) ^ 2
222 NEXT J44

223 SUM2 = 0
224 FOR J44 = 1 TO 8
225 SUM2 = SUM2 + (X(J44) + 1 / 8 ^ .5##) ^ 2

226 NEXT J44
244 REM IF -1 + EXP(-SUM2) > EPSILON2 THEN 1670

422 REM PDU = -1 + EXP(-SUM1)
431 PDU = W1 * (-1 + EXP(-SUM1)) + W2 * (-1 + EXP(-SUM2))

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 8

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

1522 g01star = -1 + EXP(-SUM1)
1524 g02star = -1 + EXP(-SUM2)
1557 GOTO 128
1670 NEXT I

1889 IF M < -.7 THEN 1999
1922 PRINT g01star, g02star, JJJJ

1924 PRINT A(1), A(2), A(3)

1926 PRINT A(4), A(5), A(6)

1928 PRINT A(7), A(8), M

1938 PRINT W1, W2

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [73]. The complete output of a single run through JJJJ= -31244 is shown below:

-.6321205588285578 -.6321205588285578 -31970
0 0 0
0 0 0
0 0 -.6321205588285578
.2  .8

-.6321205588285578 -.6321205588285578 -31656
0 0 0
0 0 0
0 0 -.6321205588285578
.8  .2

-.6321205588285578 -.6321205588285578 -31304
0 0 0
0 0 0
0 0 -.6321205588285578
.2  .8

-.6321205588285578 -.6321205588285578 -31244
0 0 0
0 0 0
0 0 -.6321205588285578
1  0
.
.
.

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 [73], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31244 was 2 seconds, not including the time for “Creating .EXE file” (17 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Siwale [66, pp. 4-7].

Acknowledgment

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

References

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

[2] 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.

[3] 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.

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

[5] 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.

[6] 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.

[7] Miguel F. Anjos (2012), FLPLIB–Facility Layout Database. Retrieved on September 25 2012 from http://www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] 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.

[9] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[10] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.

[11] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.

[12] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[13] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[14] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.

[15] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[16] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[17] Pintu Das, tapan kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, july 2014. http://www.jgrcs.info

[18] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000). Mechanical component design for multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000). (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

[19] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.

[20] Anoop K. Dhingra (1992). Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[21] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

[22] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[23] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[24] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[24] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[25] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[26] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012. Retrieved on September 14 2012 from Google search.

[27] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[28] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[29] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.

[30] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

[31] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[32] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from http://www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[33] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from http://www.optimization-online.org./DB_HTML/2012/04/3432.html

[34] Ekta Jain, Kalpana Dahiya, Vanita Verma (2018): Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[35] N. K. Jain, V. K. Jain, K. Deb (2007). Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms. International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[36] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey–Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[37] Adhe Kania, Kuntjoro Adji Sidarto (2016). Solving mixed integer nonlinear programming problems using spiral dynamics optimization algorithm. AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

[38] reena kapoor, S. R. Arora (2006). linearization of a 0-1 quadratic fractional programming problem: OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.

[39] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[40] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[41] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[42] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[43] Rein Luus (1975). Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[44] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234. Homepage: http://www.GrowingScience.com/ijiec

[45] F. Masedu, M Angelozzi (2008). Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[46] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[47] Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010). NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point. European Journal of Operational Research 206 (2010) 426-434.

[48] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.

[49] Mohamed Arezki Mellal, Edward J. Williams (2016). Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic. Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.

[50] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.

[51] 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.

[52] A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015). Multi-objective optimization of truss structures using the bee algorithm. (One can read this via Goodle search.)

[53] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[54] Subhash C. Narula, H. Roland Weistroffer (1989). A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[55] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[56] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/

[57] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[58] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

[59] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[60] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.

[61] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design. Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[62] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.

[63] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.

[64] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[65] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.

[66] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf

[67] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes’ Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

[68] Hardi Tambunan, Herman Mawengkang (2016). Solving Mixed Integer Non-Linear Programming Using Active Constraint. Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281. http://www.ripublication.com/gjpam.htm

[69] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

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34:325-334.

[72] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Joutnal of Chemical Engineering 28 (1):32-40 January 2011.

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

[74] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

Solving a Multi-Objective Integer Linear Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following integer nonlinear programming problem on p. 4 of Mehdi, Chergui, and Abas [50]:

Maximize
( (30 * X(1) + 50 * X(2) + 94 * X(3) + 94 * X(4) + 27 * X(5) + 19 * X(6) + 91 * X(7)) – 8 ) /
( (98 * X(1) + 40 * X(2) + 78 * X(3) + 56 * X(4) + 29 * X(5) + 23 * X(6) + 43 * X(7)) + 4 )

maximize
( (14 * X(1) + 46 * X(2) + 76 * X(3) + 65 * X(4) + 46 * X(5) + 7 * X(6) + 35 * X(7)) + 21 ) /
( (2 * X(1) + 18 * X(2) + 40 * X(3) + 51 * X(4) + 70 * X(5) + 98 * X(6) + 50 * X(7)) + 49 )

subject to
(21 * X(1) + 3 * X(2) + 8 * X(3) – 9 * X(4) – 1 * X(5) + 26 * X(6) + 34 * X(7)) <= 41
(33 * X(1) + 28 * X(2) + 29 * X(3) + 14 * X(4) + 9 * X(5) – 7 * X(6) – 9 * X(7)) <= 49
(28 * X(1) – 1 * X(2) + 40 * X(3) + 38 * X(4) + 28 * X(5) + 43 * X(6) + 35 * X(7)) <= 106

X(1) through X(7)>=0 and are integers.

0 REM 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

87 M = -4E+250
93 IF RND < 1 / 11 THEN W1 = 0 ELSE IF RND < 1 / 10 THEN W1 = .1 ELSE IF RND < 1 / 9 THEN W1 = .2 ELSE IF RND < 1 / 8 THEN W1 = .3 ELSE IF RND < 1 / 7 THEN W1 = .4 ELSE IF RND < 1 / 6 THEN W1 = .5 ELSE IF RND < 1 / 5 THEN W1 = .6 ELSE IF RND < 1 / 4 THEN W1 = .7 ELSE IF RND < 1 / 3 THEN W1 = .8 ELSE IF RND < 1 / 2 THEN W1 = .9 ELSE W1 = 1
99 W2 = 1 – W1

111 FOR J44 = 1 TO 7

115 A(J44) = FIX(RND * 2)
121 NEXT J44

128 FOR I = 0 TO FIX(RND * 15)
129 FOR KKQQ = 1 TO 7

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

135 FOR IPP = 1 TO FIX(1 + RND * .3)
143 j = 1 + FIX(RND * 7)

145 GOTO 162

154 IF RND < .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) ELSE X(j) = A(j) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

173 FOR J44 = 1 TO 7

174 X(J44) = INT(X(J44))
177 NEXT J44
293 FOR J44 = 1 TO 7
294 IF X(J44) < 0## THEN 1670

296 IF X(J44) > 3## THEN 1670

297 NEXT J44
333 IF (21 * X(1) + 3 * X(2) + 8 * X(3) – 9 * X(4) – 1 * X(5) + 26 * X(6) + 34 * X(7)) > 41 THEN 1670
336 IF (33 * X(1) + 28 * X(2) + 29 * X(3) + 14 * X(4) + 9 * X(5) – 7 * X(6) – 9 * X(7)) > 49 THEN 1670
339 IF (28 * X(1) – 1 * X(2) + 40 * X(3) + 38 * X(4) + 28 * X(5) + 43 * X(6) + 35 * X(7)) > 106 THEN 1670
519 UP1 = (30 * X(1) + 50 * X(2) + 94 * X(3) + 94 * X(4) + 27 * X(5) + 19 * X(6) + 91 * X(7)) – 8
521 DO1 = (98 * X(1) + 40 * X(2) + 78 * X(3) + 56 * X(4) + 29 * X(5) + 23 * X(6) + 43 * X(7)) + 4
523 UP2 = (14 * X(1) + 46 * X(2) + 76 * X(3) + 65 * X(4) + 46 * X(5) + 7 * X(6) + 35 * X(7)) + 21
525 DO2 = (2 * X(1) + 18 * X(2) + 40 * X(3) + 51 * X(4) + 70 * X(5) + 98 * X(6) + 50 * X(7)) + 49

535 P = W1 * (UP1 / DO1) + W2 * (UP2 / DO2)

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 7

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1457 STAR1 = (UP1 / DO1)
1458 STAR2 = (UP2 / DO2)

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), STAR1, STAR2, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [82]. The output of a single run through JJJJ= -31754 is summarized below:

.
.
.

0    0    0    2    0
0    0    1.551724    1    -31969

0    1    0    1    0
0    0    1.36    1.118644    -31968
.
.
.

0    0    1    1    0
0    0    1.304348    1.157143    -31960
.
.
.

0    1    0    1    0
0    1    1.587413    .9940476    -31951
.
.
.

0    0    0    0    0
0    1    1.765957    .5656565    -31949
.
.
.

0    0    0    1    0
0    1    1.718447    .8066667    -31934
.
.
.

0    2    0    0    0
0    1    1.440945    1.096296    -31754
.
.
.

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 [82], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31754 was 2 seconds, not including the time for “Creating .EXE file” (17 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Mehdi, Chergui, and Abbas [50, p. 3, Table 2 and Table 3].

Acknowledgment

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

References

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

[2] 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.

[3] 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.

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

[5] 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.

[6] 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.

[7] Miguel F. Anjos (2012), FLPLIB–Facility Layout Database. Retrieved on September 25 2012 from http://www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] 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.

[9] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

[10] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.

[11] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.

[12] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[13] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[14] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.

[15] Lino Costa, Pedro (2001). Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[16] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[17] Pintu Das, Tapan Kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, July 2014. http://www.jgrcs.info

[18] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000). Mechanical component design for multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000). (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

[19] Kusum Deep, Krishna Pratap Singh, M. L. Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation 212 (2009) 505-518.

[20] Anoop K. Dhingra (1992). Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[21] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

[22] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).

[23] C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[24] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[24] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems. Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[25] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.

[26] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012. Retrieved on September 14 2012 from Google search.

[27] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[28] Neha Gupta, Irfan Ali, Abdul Bari (2013). An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation. Journal of Mathematical Modelling and Algorithms, January 2013.

[29] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

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