Jsun Yui Wong
STAGE 1
Based on the computer program of the earlier edition, the first computer program listed below seeks to solve the following formulation, which is based on the first formulation on p. 2462 of Varshney and Mradula [86]:
Minimize
- (-.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10))
subject to
4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) <= 10000
2<= X(1) <= 395
2<= X(2) <= 382
2<= X(3) <= 439
2<= X(4) <= 368
2<= X(5) <= 416
2<= X(6) <= 30
2<= X(7) <= 30
2<= X(8) <= 30
2<= X(9) <= 30
2<= X(10) <= 30
X(1) through X(10) are general integer variables.
Here the long constraint above is handled by 591 IF 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) > 10000 THEN 1670, whereas in the earlier edition, it is handled by 594 X(1) = (-(4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4)) + 10000) / 29.
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 -31900
85 RANDOMIZE JJJJ
87 M = -4E+250
120 FOR J44 = 1 TO 5
121 A(J44) = RND * 50
122 NEXT J44
123 FOR J44 = 6 TO 10
124 A(J44) = RND * 3
125 NEXT J44
128 FOR I = 0 TO FIX(RND * 20000)
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND * 3.3)
143 j = 1 + FIX(RND * 10)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * R
161 GOTO 169
162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
171 FOR J44 = 1 TO 10
173 X(J44) = INT(X(J44))
174 IF X(J44) < 2 THEN 1670
175 NEXT J44
188 IF X(1) > 395 THEN 1670
189 IF X(2) > 382 THEN 1670
190 IF X(3) > 439 THEN 1670
191 IF X(4) > 368 THEN 1670
192 IF X(5) > 416 THEN 1670
193 IF X(6) > 30 THEN 1670
194 IF X(7) > 30 THEN 1670
195 IF X(8) > 30 THEN 1670
196 IF X(9) > 30 THEN 1670
197 IF X(10) > 30 THEN 1670
591 IF 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) > 10000 THEN 1670
594 REM X(1) = (-(4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4)) + 10000) / 29
597 REM X(1) = INT(X(1))
599 REM IF X(1) < 0 THEN 1670
1014 P = (-.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10))
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1488 REM
1489 LHS = 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4)
1557 GOTO 128
1670 NEXT I
1890 IF M < -.84411 THEN 1999
1926 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ, LHS
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [92]. The complete output of a single run through JJJJ= -31972 is shown below:
13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31993 9997.73
13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31987 9997.73
13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31972 9997.73
.
.
.
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel (R) Core (TM) i5 CPU M 430 @ 2.27 GHz, 4.00 GB of RAM, and QB64v1000-win [92], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31972 was 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Varshney and Mradula [86, p. 2462].
STAGE 2
Based on the computer program of the earlier edition, the computer program listed below seeks to solve with an active constraint the following formulation, which is based on the second formulation on p. 2462 of Varshney and Mradula [86]:
Minimize
- (-2.047828 / X(1) - 70.97196 / X(2) - 25.1802 / X(3) - 11.2677 / X(4) - 2.736015 / X(5) - .03413047 / X(6) - 2.381936 / X(7) - .6124794 / X(8) - .187795 / X(9) - .05766226 / X(10)) - X(11)
subject to
-(-.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10)) - X(11)<= .8441042055052964, where the .8441042055052964 is from stage 1
4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) <= 10000
2<= X(1) <= 395
2<= X(2) <= 382
2<= X(3) <= 439
2<= X(4) <= 368
2<= X(5) <= 416
2<= X(6) <= 30
2<= X(7) <= 30
2<= X(8) <= 30
2<= X(9) <= 30
2<= X(10) <= 30
X(1) through X(10) are general integer variables
X(11) >= 0.
One notes the following line 509, which is 509 X(11) = -(-.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10)) - .8441042055052964
and comes from the first long inequality constraint above. That is a search for an active constraint for the goal of optimization; see Bernau [10].
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 = -4E+250
120 FOR J44 = 1 TO 5
121 A(J44) = RND * 50
122 NEXT J44
123 FOR J44 = 6 TO 10
124 A(J44) = RND * 3
125 NEXT J44
126 A(11) = RND
128 FOR I = 0 TO FIX(RND * 20000)
129 FOR KKQQ = 1 TO 11
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND * 3.3)
143 j = 1 + FIX(RND * 11)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 R = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 15)) * R
161 GOTO 169
162 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
171 FOR J44 = 1 TO 10
173 X(J44) = INT(X(J44))
174 IF X(J44) < 2 THEN 1670
175 NEXT J44
177 IF X(11) < 0## THEN 1670
188 IF X(1) > 395 THEN 1670
189 IF X(2) > 382 THEN 1670
190 IF X(3) > 439 THEN 1670
191 IF X(4) > 368 THEN 1670
192 IF X(5) > 416 THEN 1670
193 IF X(6) > 30 THEN 1670
194 IF X(7) > 30 THEN 1670
195 IF X(8) > 30 THEN 1670
196 IF X(9) > 30 THEN 1670
197 IF X(10) > 30 THEN 1670
509 X(11) = -(-.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10)) - .8441042055052964
586 IF X(11) < 0## THEN 1670
591 IF 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) > 10000 THEN 1670
1016 P = (-2.047828 / X(1) - 70.97196 / X(2) - 25.1802 / X(3) - 11.2677 / X(4) - 2.736015 / X(5) - .03413047 / X(6) - 2.381936 / X(7) - .6124794 / X(8) - .187795 / X(9) - .05766226 / X(10)) - X(11)
1111 IF P <= M THEN 1670
1420 M = P
1444 FOR KLX = 1 TO 11
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1489 LHS = 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4)
1557 GOTO 128
1670 NEXT I
1890 IF M < -2.0274 THEN 1999
1926 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), A(11), M, JJJJ, LHS
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [92]. The complete output of a single run through JJJJ= -31986 is shown below:
17 92 92 39 36
2 16 14 5 5
.2380042824618931 -2.027288943816652 -31998
9999.970000000001
17 92 92 39 36
2 16 14 5 5
.2380042824618931 -2.027288943816652 -31987
9999.970000000001
17 92 92 39 36
2 16 14 5 5
.2380042824618931 -2.027288943816652 -31986
9999.970000000001
.
.
.
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with Processor Intel (R) Core (TM) i5 CPU M 430 @ 2.27 GHz, 4.00 GB of RAM, and QB64v1000-win [92], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31986 was 2 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those in Varshney and Mradulla [86, p. 2462].
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] H. Bernau (1990). Active constraint strategies in optimization. Geographical Data Inversion Methods and Applications, pp. 15-31.
[11] Victor Blanco, Justo Puerto (2011). Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.
[12] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.
[13] Regina S. Burachik, C. Yalcin Kaya, M. Mustafa Rizvi (March 19, 2019), Algorithms for Generating Pareto Fronts of Multi-objective Integer and Mixed-Integer Programming Problems, arXiv: 1903.07041v1 [math.OC] 17 Mar 2019.
[14] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[15] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[16] 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.
[17] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[18] 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.
[19] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[20] 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
[21] 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.)
[22] 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.
[23] 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.
[24] Wassila Drici, Mustapha Moulai (2019): An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.
[25] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).
[26] 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.
[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, 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.
[29] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.
[30] 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.
[31] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
[32] Neha Gupta, Irfan Ali, Abdul Bari (2013). An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation. Journal of Mathematical Modelling and Algorithms, January 2013.
[33] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[34] 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.
[35] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.
[36] 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
[37] Sana Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling design. Research Journal of Mayhematical and Statistical Sciences, vol. 3(1),10-14, January (2015).
[38] 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
[39] Golam Reza Jahanshahloo, Farhad Hosseinzadeh, Nagi Shoja, Ghasem Tohidi (2003) A method for solvong 0-1 multiple objective liner programming problem using DEA. Journal of the Operations Research Society of Japan (2003), 46 (2): 189-202. http://www.orsj.or.jp/~archive/pdf/e_mag/Vol.46_02_189.pdf
[40] 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.
[41] 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.
[42] 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.
[43] 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.
[44] 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.
[45] M. G. M. Khan, E. A. Khan, M. J. Ahsan (2003). An optimal multivariate stratified sampling design using dynamic programming. Australian and New Zealand Journal of Statistics, vol. 45, no. 1, 2003, pp. 107-113.
[46] M. G. M. Khan, T. Maiti, M. J. Ahsan (2010). An optimal multivariate stratified sampling design using auxiliary information: an integer solution using goal programming approach. Journal of Official Statistics, vol. 26, no. 4, 2010, pp. 695-708.
[47] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[48] 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.
[49] F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research 126 (2000) 51-68.
[50] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[51] 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.
[52] 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.
[53] 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
[54] 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.
[55] 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.
[56] 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.
[57] 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.
[58] 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.
[59] 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.)
[60] 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.
[61] 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.
[62] 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.
[63] A. K. Ojha, K. K. Biswal (2010). Multi-objective geometric programming problem with weighted mean method. (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp. 82-86, 2010.
[64] Rashmi Ranjan Ota, jayanta kumar dASH, A. K. Ojha (2014). A hybrid optimization techniques for solving non-linear multi-objective optimization problem. amo-advanced modelling and optimization, volume 16,number 1, 2014.
[65] 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/
[66] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system. OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.
[67] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[68] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[69] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.
[70] Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statitiscal Computation ans Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370.
[71] 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.
[72] 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.
[73] 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.
[74] Shafiullah, Irfan Ali, Abdul Bari (2015). Fuzzy geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp. 021-035 (2015).
[75] 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.
[76] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[77] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[78] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf
[79] 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.
[80] Har3i 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
[81] 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.
[82] 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.
[83] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[84] 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.
[85] 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.
[86] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.
[87] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.
[88] 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.
[89] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017
[90] 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.
[91] 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.
[92] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[93] 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.
