Solving Multi-Objective Mixed-Integer Nonlinear Programs by Example

Jsun Yui Wong

The problem for the computer program listed below is based on the (following) last formulation on page 193 of Ali, Raghav, and Bari [2]:

Minimize      delta1+delta2

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)))) -delta1= 63.90

(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)))) -delta2= 108.75

(1 – (1 – .9) ^ (1 + X(1))) * (1 – (1 – .85) ^ (2 + X(2))) * (1 – (1 – .85) ^ (2 + X(3))) * (1 – (1 – .8) ^ (2 + X(4))) * (1 – (1 – .85) ^ (1 + X(5))) >= .99

delta1 >=0

delta2 >=0

0<= X(1) <=3
0<= X(2) <= 2
0<= X(3) <= 2
0<= X(4) <= 2
0<= X(5) <= 3

X(1) through X(5) are integer variables.
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 = -3E+50
116 FOR J44 = 1 TO 7
120 A(J44) = RND * 3
121 NEXT J44
128 FOR I = 1 TO FIX(6000 + RND * 6000)
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 4)
143 J = 1 + FIX(RND * 5)
149 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)
169 NEXT IPP
170 FOR J44 = 1 TO 5

171 X(J44) = INT(X(J44))
172 IF X(J44) < 0 THEN 1670
174 NEXT J44
176 IF X(1) > 3 THEN 1670
177 IF X(2) > 2 THEN 1670
178 IF X(3) > 2 THEN 1670
179 IF X(4) > 2 THEN 1670
180 IF X(5) > 3 THEN 1670

409 X(6) = ABS((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)))) – 63.90)
417 X(7) = ABS((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)))) – 108.75)
431 IF (1 – (1 – .9) ^ (1 + X(1))) * (1 – (1 – .85) ^ (2 + X(2))) * (1 – (1 – .85) ^ (2 + X(3))) * (1 – (1 – .8) ^ (2 + X(4))) * (1 – (1 – .85) ^ (1 + X(5))) < .99 THEN 1670
438 PDU = -X(6) – X(7)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 7

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

1557 GOTO 128
1670 NEXT I
1890 IF M < -9999999 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), A(6), A(7), M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [79]. The complete output of a single run through JJJJ= -31994 is showm below:

2      1       1       2
3       1.745948       2.963402E-03       -1.748912
-32000

2       1       1       2
3       1.745948       2.963402E-03       -1.748912
-31996

2       1       1       2
3       1.745948       2.963402E-03       -1.748912
-31995

2       1       1       2
3       1.745948       2.963402E-03       -1.748912
-31994

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 [79], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 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 on page 193 of Ali, Raghav, and Bari [2].

Acknowledgment

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

References

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Solving Multi-Objective Integer Nonlinear Programs by Example

Jsun Yui Wong

The computer program listed below seeks to solve the following integer programming formulation (that uses the weight criterion approach) from page 122 of Ali and Hasan [1]:

Maximize       .47*long1+.53*log2

where long1 = (1 – .35 ^ (4 + X(1))) * (1 – .45 ^ (2 + X(2))) * (1 – .3 ^ (3 + X(3)))

long2 = (1 – .3 ^ (3 + X(4))) * (1 – .45 ^ (2 + X(5))) * (1 – .4 ^ (2 + X(6))) * (1 – .35 ^ (3 + X(7)))

subject to

140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) <= 3000

1<= X(1) <=4
1<= X(2) <= 4
1<= X(3) <= 7
1<= X(4) <= 5
1<= X(5) <= 8
1<= X(6) <= 10
1<= X(7) <= 7

X(1) through X(7) are integer variables.
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 = -3E+50

111 REM EPSILON = RND * .5

113 FOR J44 = 1 TO 7

120 A(J44) = 1 + FIX((RND * 9))

125 NEXT J44

128 FOR I = 1 TO FIX(6000 + RND * 6000)
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 J = 1 + FIX(RND * 7)

154 IF RND < .5 THEN 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) ELSE X(J) = A(J) + FIX(1 + RND * 2)
169 NEXT IPP
170 FOR J44 = 1 TO 7

171 X(J44) = INT(X(J44))
172 IF X(J44) < 1 THEN 1670
173 NEXT J44

174 IF X(1) > 4 THEN 1670
175 IF X(2) > 4 THEN 1670
176 IF X(3) > 7 THEN 1670
177 IF X(4) > 5 THEN 1670
178 IF X(5) > 8 THEN 1670
179 IF X(6) > 10 THEN 1670
180 IF X(7) > 7 THEN 1670

411 IF 140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) > 3000 THEN 1670
423 long1 = (1 – .35 ^ (4 + X(1))) * (1 – .45 ^ (2 + X(2))) * (1 – .3 ^ (3 + X(3)))

425 long2 = (1 – .3 ^ (3 + X(4))) * (1 – .45 ^ (2 + X(5))) * (1 – .4 ^ (2 + X(6))) * (1 – .35 ^ (3 + X(7)))
459 PDU = (.47 * long1 + .53 * long2)
466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 7

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -99999999 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4), A(5)
1924 PRINT A(6), A(7), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [79]. The complete output of a single run through JJJJ= -31978 is showm below:

1       4       2       3       6
5       3       .989409       -31997

1       4       2       3       6
5       3      .989409       -31990

1       4       2       3       6
5       3       .989409       -31983

1       4       2       3       6
5       3      .989409       -31981

1       4       2       3       6
5       3       .989409       -31978

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 [79], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31978 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 on page 122 of Ali and Hasan [1].

Acknowledgment

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

References

[1] Irfan Ali, S. Suhaib Hasan (2014). Integer fuzzy programming approach in bi-objective selective maintenance allocation problem. Journal of  Mathematical  Modelling and  Algorithms in Operations Research (2014) 13:113-124.

[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, Vol. 3, No. 6, 2011, pp. 184-195.

[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.

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

[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.

[11] 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.

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

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

[14] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
.
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[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
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[33] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.

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

[35] 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

[36] 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

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Solving Multi-Objective Integer Nonlinear Programs by Example

Jsun Yui Wong

The computer program listed below seeks to solve the following integer programming formulation (that uses the desirability function approach) from page 121 of Ali and Hasan [1]:

Maximize           ((ratio1 ^ .47) * (ratio2 ^ .53)) ^ .5

where ratio1 = (long1 – .8967782) / (.0940632)

ratio2 = (long2 – .8311538) / (.1678001)

long1 = (1 – .35 ^ (4 + X(1))) * (1 – .45 ^ (2 + X(2))) * (1 – .3 ^ (3 + X(3)))

long2 = (1 – .3 ^ (3 + X(4))) * (1 – .45 ^ (2 + X(5))) * (1 – .4 ^ (2 + X(6))) * (1 – .35 ^ (3 + X(7)))

subject to

140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) <= 3000

1<= X(1) <=4
1<= X(2) <= 4
1<= X(3) <= 7
1<= X(4) <= 5
1<= X(5) <= 8
1<= X(6) <= 10
1<= X(7) <= 7

X(1) through X(7) are integer variables.

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 = -3E+50

111 REM EPSILON = RND * .5

113 FOR J44 = 1 TO 7

120 A(J44) = 1 + FIX((RND * 7))

125 NEXT J44

128 FOR I = 1 TO FIX(6000 + RND * 6000)
129 FOR KKQQ = 1 TO 7
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 J = 1 + FIX(RND * 7)

154 IF RND < .5 THEN 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) ELSE X(J) = A(J) + FIX(1 + RND * 2)
169 NEXT IPP
170 FOR J44 = 1 TO 7

171 X(J44) = INT(X(J44))
172 IF X(J44) < 1 THEN 1670
173 NEXT J44

174 IF X(1) > 4 THEN 1670
175 IF X(2) > 4 THEN 1670
176 IF X(3) > 7 THEN 1670
177 IF X(4) > 5 THEN 1670
178 IF X(5) > 8 THEN 1670
179 IF X(6) > 10 THEN 1670
180 IF X(7) > 7 THEN 1670

411 IF 140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) > 3000 THEN 1670
423 long1 = (1 – .35 ^ (4 + X(1))) * (1 – .45 ^ (2 + X(2))) * (1 – .3 ^ (3 + X(3)))

425 long2 = (1 – .3 ^ (3 + X(4))) * (1 – .45 ^ (2 + X(5))) * (1 – .4 ^ (2 + X(6))) * (1 – .35 ^ (3 + X(7)))

434 ratio1 = (long1 – .8967782) / (.0940632)

437 ratio2 = (long2 – .8311538) / (.1678001)
457 PDU = ((ratio1 ^ .47) * (ratio2 ^ .53)) ^ .5
466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 7

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -99999999 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4), A(5)
1924 PRINT A(6), A(7), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [79]. The complete output of a single run through JJJJ= -31993 is showm below:

2       4       2       2       6
5       2       .9755799       -31997

2       4       2       2       6
5       2       .9755799       -31996

2       4       2       2       6
5       2       .9755799       -31995

2       4       2       2       6
5       2       .9755799       -31994

1       4       2       3       6
5       3       .9750768       -31993

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 [79], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31993 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 on page 121 of Ali and Hasan [1].

Acknowledgment

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

References

[1] Irfan Ali, S. Suhaib Hasan (2014). Integer fuzzy programming approach in bi-objective selective maintenance allocation problem. Journal of Mathematical Modelling and Algorithms in Operations Research (2014) 13:113-124.

[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, Vol. 3, No. 6, 2011, pp. 184-195.

[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.

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

[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.

[11] 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.

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

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

[14] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
.
[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[32] 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.

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

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

[35] 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

[36] 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

[37] 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.

[38] 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.

[39] 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.

[40] 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.

[41] 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.

[42] Mohammed Faisal Khan, Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2012) Allocation in multivariate stratified surveys with non-linear random cost function. American Journal of operations research, 2012, 2, 100-105.

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

[44] 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.

[45] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[46] 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.

[47] 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.

[48] 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

[49] 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.

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

[51] 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.

[52] 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.

[53] 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.

[54] 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.

[55] 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.

[56] 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.)

[57] 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.

[58] 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.

[59] 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.

[60] 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/

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

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

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

[64] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36.

[65] 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.

[66] 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.

[67] 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.

[68] 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.

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

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Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

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[74] 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.

[75] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[76] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys. Annals of Operations Research (2015) 226:659-668.

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34:325-334.

[78] 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.

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

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Solving Multi-Objective Integer Nonlinear Programs by Example

Jsun Yui Wong

The computer program listed below seeks to solve the last problem in Ali, Raghav, and Bari [1; p. 194]:

Minimize      (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))))

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)))) <= 63.90

(1 – (1 – .9) ^ (1 + X(1))) * (1 – (1 – .85) ^ (2 + X(2))) * (1 – (1 – .85) ^ (2 + X(3))) * (1 – (1 – .8) ^ (2 + X(4))) * (1 – (1 – .85) ^ (1 + X(5))) >= .99

0<= X(1) <= 3
0<= X(2) <= 2
0<= X(3) <= 2
0<= X(4) <= 2
0<= X(5) <= 3

X(1) through X(5) are integer variables.
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
120 A(1) = FIX((RND * 3))
121 A(2) = FIX((RND * 3))
122 A(3) = FIX((RND * 3))
123 A(4) = FIX((RND * 3))

124 A(5) = FIX((RND * 3))
128 FOR I = 1 TO FIX(2000 + RND * 2000)
129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 3)
143 J = 1 + FIX(RND * 5)

146 IF J < 5.5 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 * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)
169 NEXT IPP
170 FOR J44 = 1 TO 5

171 X(J44) = INT(X(J44))
172 IF X(J44) < 0 THEN 1670
174 NEXT J44
176 IF X(1) > 3 THEN 1670
177 IF X(2) > 2 THEN 1670
178 IF X(3) > 2 THEN 1670
179 IF X(4) > 2 THEN 1670
180 IF X(5) > 3 THEN 1670
409 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)))) > 63.90 THEN 1670
431 IF (1 – (1 – .9) ^ (1 + X(1))) * (1 – (1 – .85) ^ (2 + X(2))) * (1 – (1 – .85) ^ (2 + X(3))) * (1 – (1 – .8) ^ (2 + X(4))) * (1 – (1 – .85) ^ (1 + X(5))) < .99 THEN 1670

440 PDU = -(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))))

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 5

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

1557 GOTO 128
1670 NEXT I
1890 IF M < -99999 THEN 1999
1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [78]. The complete output of a single run through JJJJ= -31994 is showm below:

2       1       2       2
2       -110.8549309550175       -32000

2       1       2       2
2       -110.8549309550175       -31999

2       1       2       2
2       -110.8549309550175       -31996

2       1       2       2
2       -110.8549309550175       -31994

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 = -31994 was 2 seconds, not including the time for “Creating .EXE file” (15 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those of the last problem in Ali, Raghav, and Bari [1; p. 194].
Acknowledgment

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

References

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[34] 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

[35] 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

[36] 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.

[37] 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.

[38] 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.

[39] 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.

[40] 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.

[41] Mohammed Faisal Khan, Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2012) Allocation in multivariate stratified surveys with non-linear random cost function. American Journal of operations research, 2012, 2, 100-105.

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

[43] 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.

[44] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[45] 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.

[46] 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.

[47] 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

[48] 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.

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

[50] 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.

[51] 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.

[52] 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.

[53] 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.

[54] 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.

[55] 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.)

[56] 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.

[57] 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.

[58] 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.

[59] 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/

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

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

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

[63] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36.

[64] 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.

[65] 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.

[66] 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.

[67] 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.

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

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

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

[71] 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.

[72] 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

[73] 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.

[74] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[75] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys. Annals of Operations Research (2015) 226:659-668.

[76] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems. ZOR – Methods and Models of Operations Research (1990)
34:325-334.

[77] 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.

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

[79] 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 Multi-Objective Mixed-Integer Nonlinear Programs in Multivariate Stratified Sampling by Example

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from pages 32-33 of Raghav, Ali, and Bari [61]:

Minimize      X(9)+ X(10)

subject to

411 X(9) = ((.7893 + (X(5) – 1) * .06) / X(1) + (.4411 + (X(6) – 1) * .02) / X(2) + (.3576 + (X(7) – 1) * .02) / X(3) + (.3973 + (X(8) – 1) * .03) / X(4)) – X(9) <=.00655

415 X(10) = ((.8316 + (X(5) – 1) * .06) / X(1) + (.3358 + (X(6) – 1) * .02) / X(2) + (.1062 + (X(7) – 1) * .01) / X(3) + (.2791 + (X(8) – 1) * .02) / X(4)) – X(10)<= .00454

((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

2<= X(1) <= 1214
2<= X(2) <= 822
2<= X(3) <= 1028
2<= X(4) <= 786.

X(1) through X(4) are integer variables, and X(5) through X(8) are continuous and >1.

The following computer program tries the what if analysis of what if the three (long) constraints above are binding–please see lines 402, 411, and 415:
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
116 A(9) = ((RND * 2))
117 A(10) = ((RND * 2))

120 A(1) = 2 + ((RND * 500))
121 A(2) = 2 + ((RND * 500))
122 A(3) = 2 + ((RND * 500))
123 A(4) = 2 + ((RND * 500))

124 A(5) = 1 + ((RND * 3))
125 A(6) = 1 + ((RND * 3))

126 A(7) = 1 + ((RND * 3))
127 A(8) = 1 + ((RND * 3))

128 FOR I = 1 TO FIX(120000 + RND * 120000)
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

135 FOR IPP = 1 TO FIX(1 + RND * 4)
143 J = 1 + FIX(RND * 10)

146 IF J < 4.5 THEN GOTO 149 ELSE GOTO 156
149 IF RND < .75 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) ELSE X(J) = A(J) + FIX(1 + RND * 3)
169 NEXT IPP
170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 1214 THEN 1670
175 IF X(2) > 822 THEN 1670
176 IF X(3) > 1028 THEN 1670
177 IF X(4) > 786 THEN 1670
190 FOR J44 = 5 TO 8

192 IF X(J44) <= 1 THEN 1670
193 NEXT J44

197 IF X(9) < 0 THEN 1670

198 IF X(10) < 0 THEN 1670
402 X(5) = -.9 * X(1) / ((2.4 * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) – 5000)

403 IF X(5) < 1 THEN 1670
411 X(9) = ((.7893 + (X(5) – 1) * .06) / X(1) + (.4411 + (X(6) – 1) * .02) / X(2) + (.3576 + (X(7) – 1) * .02) / X(3) + (.3973 + (X(8) – 1) * .03) / X(4)) – .00655

413 IF X(9) < 0 THEN 1670
415 X(10) = ((.8316 + (X(5) – 1) * .06) / X(1) + (.3358 + (X(6) – 1) * .02) / X(2) + (.1062 + (X(7) – 1) * .01) / X(3) + (.2791 + (X(8) – 1) * .02) / X(4)) – .00454
416 IF X(10) < 0 THEN 1670
438 PDU = -X(9) – X(10)
466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 10

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -.00011970 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [76]. The complete output of a single run through JJJJ= -31997 is showm below:

528       311       220       246
2.15873363446326       2.07639467734374       2.120076657318881
2.128821577989015       4.40916517890466D-05       7.560537019565681D-05
1.196970219847034D-04       -32000

529       311      220       246
2.170679345033012       2.083929209761695       2.127769701299336
2.136546329060663       4.449669448031563D-05       7.519527125169796D-05
1.196919657320136D-04       -31999

529       311       220       246
2.170670341453166       2.083929126162365       2.127769735516927
2.13654636299655          4.44966959478776D-05       7.519526978413577D-05
1.196919657320134D-04       -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 [76], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 4 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 on page 33 of Raghav, Ali, and Bari [61].
Acknowledgment

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

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

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Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

[70] 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

[71] 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.

[72] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

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[74] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems. ZOR – Methods and Models of Operations Research (1990)
34:325-334.

[75] 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.

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

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Solving Another Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from page 32 of Raghav, Ali, and Bari [61; the second problem on p. 32]:

Minimize ((.8316 + (X(5) – 1) * .06) / X(1) + (.3358 + (X(6) – 1) * .02) / X(2) + (.1062 + (X(7) – 1) * .01) / X(3) + (.2791 + (X(8) – 1) * .02) / X(4))

subject to

((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

2<= X(1) <= 1214
2<= X(2) <= 822
2<= X(3) <= 1028
2<= X(4) <= 786.

X(1) through X(4) are integer variables, and X(5) through X(8) are continuous and >1.

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

120 A(1) = 2 + ((RND * 500))
121 A(2) = 2 + ((RND * 500))
122 A(3) = 2 + ((RND * 500))
123 A(4) = 2 + ((RND * 500))

124 A(5) = 1 + ((RND * 3))
125 A(6) = 1 + ((RND * 3))

126 A(7) = 1 + ((RND * 3))
127 A(8) = 1 + ((RND * 3))

128 FOR I = 1 TO FIX(120000 + RND * 120000)

129 FOR KKQQ = 1 TO 8

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

151 FOR IPP = 1 TO FIX(1 + RND * 4)
153 J = 1 + FIX(RND * 8)

154 IF RND < .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 * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)
169 NEXT IPP
170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 1214 THEN 1670
175 IF X(2) > 822 THEN 1670
176 IF X(3) > 1028 THEN 1670
177 IF X(4) > 786 THEN 1670
190 FOR J44 = 5 TO 8

192 IF X(J44) <= 1 THEN 1670
193 NEXT J44
423 IF ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) > 5000 THEN 1670
444 PDU = -((.8316 + (X(5) – 1) * .06) / X(1) + (.3358 + (X(6) – 1) * .02) / X(2) + (.1062 + (X(7) – 1) * .01) / X(3) + (.2791 + (X(8) – 1) * .02) / X(4))
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

1557 GOTO 128
1670 NEXT I

1889 IF M < -.007 THEN 1999
1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

.
This BASIC computer program was run with QB64v1000-win [76]. The output of a single run through JJJJ= -31997 is summarized below:

.
.
.
593       321       163       249
2.177165922925384       1.926564694447423       1.744560694620744
2.169946237545697      -4.537370279648064D-03       -31998
595       321       163       249
2.208299520140577       1.943336232570953       1.729004715705757
2.19337853751922       -4.53736834068757D-03       -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 [76], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 7 seconds, not including the time for “Creating .EXE file” (20 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 32 of Raghav, Ali, and Bari [61].
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.

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[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.
.
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[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.

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[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

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[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).
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[41] 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.

[42] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
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[44] 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.

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[46] 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.

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[49] 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.

[50] 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.

[51] 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.

[52] 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.

[53] 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.)

[54] 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.

[55] 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.

[56] 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.

[57] 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/

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

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

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

[61] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36.

[62] 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.

[63] 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.

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

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Solving an Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The present paper considers the integer version of the following problem from page 103 of Khan, Raghav, Ali, and Bari [39]:

Minimize 11333.5688 / X(1) + 158.6615 / X(2) + 166.1824 / X(3) + 2960.5328 / X(4)

subject to

(3 * X(1) ^ .5 + 4 * X(2) ^ .5 + 5 * X(3) ^ .5 + 7 * X(4) ^ .5) + 2.33 * ((.6 * X(1) + .5 * X(2) + .7 * X(3) + .8 * X(4)) ^ .5) <= 275

2<= X(1) <= 1419
2<= X(2) <= 619
2<= X(3) <= 1253
2<= X(4) <= 899.

One notes line 171, which is 171 X(J44) = INT(X(J44)).

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

111 REM EPSILON = RND * .5

120 A(1) = 2 + FIX((RND * 500))
121 A(2) = 2 + FIX((RND * 500))
122 A(3) = 2 + FIX((RND * 500))
123 A(4) = 2 + FIX((RND * 500))
128 FOR I = 1 TO FIX(10000 + RND * 10000)

129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 4)

154 IF RND < .5 THEN 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)
169 NEXT IPP
170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 1419 THEN 1670
175 IF X(2) > 619 THEN 1670
176 IF X(3) > 1253 THEN 1670
177 IF X(4) > 899 THEN 1670

406 IF (3 * X(1) ^ .5 + 4 * X(2) ^ .5 + 5 * X(3) ^ .5 + 7 * X(4) ^ .5) + 2.33 * ((.6 * X(1) + .5 * X(2) + .7 * X(3) + .8 * X(4)) ^ .5) > 275 THEN 1670

450 PDU = -11333.5688 / X(1) – 158.6615 / X(2) – 166.1824 / X(3) – 2960.5328 / X(4)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 4

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -44.581 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [76]. The complete output of a single run through JJJJ= -31839 is shown below:

623       37       34       172       -44.58019053326766
-31996
623 37 34 172 -44.58019053326766
-31995
623 37 34 172 -44.58019053326766
-31962
618 37 33 175 -44.58041732800812
-31952
618 37 33 175 -44.58041732800812
-31940
618 37 33 175 -44.58041732800812
-31939
618 37 33 175 -44.58041732800812
-31934
623 37 34 172 -44.58019053326766
-31897
623 37 34 172 -44.58019053326766
-31893
618 37 33 175 -44.58041732800812
-31888
618 37 33 175 -44.58041732800812
-31878
618 37 33 175 -44.58041732800812
-31853
623 37 34 172 -44.58019053326766
-31840
618 37 33 175 -44.58041732800812
-31839

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 [76], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31839 was 13 seconds, not including the time for “Creating .EXE file” (25 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 104 of Khan, Raghav, Ali, and Bari [39].

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] Mohammed Faisal Khan, Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2012) Allocation in multivariate stratified surveys with non-linear random cost function. American Journal of  Operations Research, 2012, 2, 100-105.

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

[41] 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.

[42] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[43] 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.

[44] 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.

[45] 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

[46] 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.

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

[48] 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.

[49] 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.

[50] 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.

[51] 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.

[52] 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.

[53] 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.)

[54] 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.

[55] 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.

[56] 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.

[57] 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/

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

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

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

[61] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36.

[62] 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.

[63] 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.

[64] 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.

[65] 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.

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

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

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

[69] 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.

[70] 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

[71] 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.

[72] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[73] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys. Annals of Operations Research (2015) 226:659-668.

[74] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems. ZOR – Methods and Models of Operations Research (1990)
34:325-334.

[75] 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.

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

[77] 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 Mixed-Integer Nonlinear Programming Problem in Multivariate Stratified Sampling Using Goal Programming

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from p. 665 of Varshney, Khan, Fatma, and Ahsan [72]:

Minimize    X(5) + X(6)

subject to

(.000847142 / X(1) + .013634624 / X(2) + .038797187 / X(3) + .002001201 / X(4)) ^ .5 – X(5) <= .0697399

(.002042574 / X(1) + .236781300 / X(2) + .089354235 / X(3) + .002296662 / X(4)) ^ .5 – X(6) <= .157701

12 * X(1) + 9 * X(2) + 10 * X(3) + 8 * X(4) <= 300

2<= X(1) <= 8
2<= X(2) <= 34
2<= X(3) <= 45
2<= X(4) <= 12

X(1) through X(4) are integers and X(5) through X(6) are continuous and >=0.

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

111 REM EPSILON = RND * .5

116 A(5) = ((RND))
117 A(6) = ((RND))

120 A(1) = 2 + ((RND * 5))
121 A(2) = 2 + ((RND * 5))
122 A(3) = 2 + ((RND * 5))
123 A(4) = 2 + ((RND * 5))

128 FOR I = 1 TO FIX(1000 + RND * 1000)

129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 2)

153 J = 1 + FIX(RND * 6)

154 IF J < 4.5 THEN 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 * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)
169 NEXT IPP

170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 8 THEN 1670
175 IF X(2) > 34 THEN 1670
176 IF X(3) > 45 THEN 1670
177 IF X(4) > 12 THEN 1670

197 IF X(5) < 0 THEN 1670

198 IF X(6) < 0 THEN 1670

406 IF (.000847142 / X(1) + .013634624 / X(2) + .038797187 / X(3) + .002001201 / X(4)) ^ .5## – X(5) > .0697399 THEN 1670

419 IF (.002042574 / X(1) + .236781300 / X(2) + .089354235 / X(3) + .002296662 / X(4)) ^ .5## – X(6) > .157701 THEN 1670

422 IF 12 * X(1) + 9 * X(2) + 10 * X(3) + 8 * X(4) > 300 THEN 1670
438 PDU = -X(5) – X(6)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 6

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -.0066 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1924 REM PRINT A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [75]. The complete output of a single run through JJJJ= -31317 is shown below:

2       15       12       2       4.86736985449209d-03
1.676607250828967d-03       -6.543977105321055D-03       -31991

2       15       12       2       4.867369854673986d-03
1.676607250898897d-03       -6.543977105572883D-03       -31919

2       15       12       2       4.86736985452615d-03
1.676607250829264d-03       -6.543977105355415D-03       -31317

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 [75], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31317 was 27 seconds, not including the time for “Creating .EXE file” (38 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 665 of Varshney, Khan, Fatima, and Ahsan [72].

Acknowledgment

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

References

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[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] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36.

[61] 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.

[62] 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.

[63] 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.

[64] 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.

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

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

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

[68] 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.

[69] 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

[70] 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.

[71] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[72] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys.  Annals of Operations Research (2015) 226:659-668.

[73] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems. ZOR – Methods and Models of Operations Research (1990)
34:325-334.

[74] 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.

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

[76] 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 Another Multi-Objective Mixed-Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The computer program listed below seeks to solve the following problem based on and different from the problem on p. 34 of Raghav, Ali, and Bari [60, p. 34, with the epsilon-constraint technique]:

Minimize (1 / 611.57) * ((493.33 + (X(5) – 1) * 37) / X(1) + (275.68 + (X(6) – 1) * 13.78) / X(2) + (223.52 + (X(7) – 1) * 13.97) / X(3) + (248.29 + (X(8) – 1) * 17.38) / X(4))

subject to

(1 / 994.14) * ((831.61 + (X(5) – 1) * 62.37) / X(1) + (335.76 + (X(6) – 1) * 16.79) / X(2) + (106.17 + (X(7) – 1) * 6.64) / X(3) + (279.11 + (X(8) – 1) * 19.54) / X(4)) >= .004537

((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

2<= X(1) <= 1214
2<= X(2) <= 822
2<= X(3) <= 1028
2< X(4) <= 786

X(1) through X(4) are integers and X(5) through X(8) are continuous and >1.

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 = -3E+50

111 REM EPSILON = RND * .5
120 A(1) = 2 + ((RND * 500))
121 A(2) = 2 + ((RND * 500))
122 A(3) = 2 + ((RND * 500))
123 A(4) = 2 + ((RND * 500))

124 A(5) = 1 + ((RND * 5))
125 A(6) = 1 + ((RND * 5))

126 A(7) = 1 + ((RND * 5))
127 A(8) = 1 + ((RND * 5))
128 FOR I = 1 TO FIX(50000 + RND * 60000)

129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 4)
153 J = 1 + FIX(RND * 8)

154 IF J < 4.5 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 * 0) ELSE X(J) = A(J) + FIX(1 + RND * 0)
169 NEXT IPP
170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 1214 THEN 1670
175 IF X(2) > 822 THEN 1670
176 IF X(3) > 1028 THEN 1670
177 IF X(4) > 786 THEN 1670
190 FOR J44 = 5 TO 8

192 IF X(J44) <= 1 THEN 1670
193 NEXT J44
427 REM IF (1 / 994.14) * ((831.61 + (X(5) – 1) * 62.37) / X(1) + (335.76 + (X(6) – 1) * 16.79) / X(2) + (106.17 + (X(7) – 1) * 6.64) / X(3) + (279.11 + (X(8) – 1) * 19.54) / X(4)) < EPSILON THEN 1670
429 IF (1 / 994.14) * ((831.61 + (X(5) – 1) * 62.37) / X(1) + (335.76 + (X(6) – 1) * 16.79) / X(2) + (106.17 + (X(7) – 1) * 6.64) / X(3) + (279.11 + (X(8) – 1) * 19.54) / X(4)) < .004537 THEN 1670
431 IF ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) > 5000 THEN 1670

436 PDU = -(1 / 611.57) * ((493.33 + (X(5) – 1) * 37) / X(1) + (275.68 + (X(6) – 1) * 13.78) / X(2) + (223.52 + (X(7) – 1) * 13.97) / X(3) + (248.29 + (X(8) – 1) * 17.38) / X(4))
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

1557 GOTO 128
1670 NEXT I

1889 IF M < -.0067016 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [74]. The complete output of a single run through JJJJ= -31998 is shown below:

480       306      253       248
2.130052       2.108157       2.165849       2.194564       -6.701499E-03
-32000

481       307       253       247
2.15241       2.10947      2.142579       2.202004        -6.70149E-03
-31999

480     307      253      248
2.139407       2.129776       2.1756       2.206835       -6.701487E-03
-31998

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 [74], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 5 seconds, not including the time for “Creating .EXE file” (16 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 Another Multi-Objective Mixed-Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The computer program listed below seeks to solve the second last problem in Raghav, Ali, and Bari [60, p.34, first of two distance-based techniques]:

Minimize ((1 / 611.57) * ((493.33 + (X(5) – 1) * 37) / X(1) + (275.68 + (X(6) – 1) * 13.78) / X(2) + (223.52 + (X(7) – 1) * 13.97) / X(3) + (248.29 + (X(8) – 1) * 17.38) / X(4)) – .006701) ^ 2 + ((1 / 994.14) * ((831.61 + (X(5) – 1) * 62.37) / X(1) + (335.76 + (X(6) – 1) * 16.79) / X(2) + (106.17 + (X(7) – 1) * 6.64) / X(3) + (279.11 + (X(8) – 1) * 19.54) / X(4)) – .004537) ^ 2

subject to

((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

2<= X(1) <= 1214
2<= X(2) <= 822
2<= X(3) <= 1028
2< X(4) <= 786

X(1) through X(4) are integers and X(5) through X(8) are continuous and >1.
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 = -3E+50
120 A(1) = 100 + ((RND * 500))
121 A(2) = 100 + ((RND * 500))
122 A(3) = 100 + ((RND * 500))
123 A(4) = 100 + ((RND * 500))

124 A(5) = 1 + ((RND * 2))
125 A(6) = 1 + ((RND * 2))

126 A(7) = 1 + ((RND * 2))
127 A(8) = 1 + ((RND * 2))
128 FOR I = 1 TO FIX(50000 + RND * 60000)

129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 4)
153 J = 1 + FIX(RND * 8)

154 IF J < 4.5 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 * 0) ELSE X(J) = A(J) + FIX(1 + RND * 0)
169 NEXT IPP
170 FOR J44 = 1 TO 4

171 X(J44) = INT(X(J44))
172 IF X(J44) < 2 THEN 1670
173 NEXT J44

174 IF X(1) > 1214 THEN 1670
175 IF X(2) > 822 THEN 1670
176 IF X(3) > 1028 THEN 1670
177 IF X(4) > 786 THEN 1670

190 FOR J44 = 5 TO 8
192 IF X(J44) <= 1 THEN 1670
193 NEXT J44

433 IF ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) > 5000 THEN 1670
436 PDU = -((1 / 611.57) * ((493.33 + (X(5) – 1) * 37) / X(1) + (275.68 + (X(6) – 1) * 13.78) / X(2) + (223.52 + (X(7) – 1) * 13.97) / X(3) + (248.29 + (X(8) – 1) * 17.38) / X(4)) – .006701) ^ 2 – ((1 / 994.14) * ((831.61 + (X(5) – 1) * 62.37) / X(1) + (335.76 + (X(6) – 1) * 16.79) / X(2) + (106.17 + (X(7) – 1) * 6.64) / X(3) + (279.11 + (X(8) – 1) * 19.54) / X(4)) – .004537) ^ 2
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

1557 GOTO 128
1670 NEXT I

1889 IF M < -8.041E-09 THEN 1999

1923 PRINT A(1), A(2), A(3), A(4)

1924 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [74]. The complete output of a single run through JJJJ= -31525 is shown below:

534       312       217       247
2.162752       2.118402       2.168251       2.204213       -8.040544E-09
-31687

533       312       217       247
2.136629       2.113473       2.174246       2.202173       -8.040884E-09
-31630

533       312       217       247
2.143074       2.116413       2.171713       2.194095       -8.039911E-09
-31525

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 [74], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31525 was 140 seconds, not including the time for “Creating .EXE file” (155 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 34 of Raghav, Ali, and Bari [60].

Acknowledgment

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

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