"What If..." and
Discrete Variables To Help Solve Nonlinear Programming Problems
Jsun Yui Wong
Similar to the computer
program of the preceding paper, the computer program listed below seeks to
solve the following mathematical formulation in Wang, Zhang, and Gao [99, p.
1514, Example 2], which is as follows:
Minimize
X(1)
subject to
X(1) ^ -1 * X(3) ^ -1 * X(4) ^ -1 * X(6) + 5
* X(1) ^ -1 * X(2) ^ .5 * X(5) * X(6) <= 1
X(3) ^ .3333333 * X(4) - X(5) ^ .5
<= -1
-X(6) - 2 * X(1) * X(2) * X(3) * X(4)
^ 4 * X(5) ^ -1 * X(6) <= -1
30<= X(1) <= 40
.01<= X(2) <= 1
.0001<= X(3) <= 1
15<= X(4) <= 20
15<= X(5) <= 20
.1 <= X(6) <= 1.
0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(200),
H(99), L(99), U(99), X(200), D(111), P(111), PS(33), J(99), AA(99), HR(32),
HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)
79 FOR JJJJ = -32000 TO
32000
80 RANDOMIZE JJJJ
90 M = -3D+30
95 A(1) = INT(100 * (30 + RND * 10)) / 100
97 A(2) = INT(100 * (.01 + RND * .99)) /
100
98 A(3) = INT(100 * (.0001 + RND * .9999))
/ 100
99 A(4) = INT(100 * (15 + RND * 5)) / 100
100 A(5) = INT(100 * (15 + RND * 5)) / 100
101 A(6) = INT(100 * (.1 + RND * .9)) / 100
123 FOR I = 1 TO 10000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO FIX(1 + RND * 3)
140 B = 1 + FIX(RND * 6)
144 IF RND < .5 THEN 160 ELSE
GOTO 166
160 r = (1 - RND * 2) * A(B)
164 X(B) = A(B) + FIX((RND ^ (RND *
15)) * r)
165 GOTO 168
166 IF RND < .5 THEN X(B) = A(B)
- FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)
168 NEXT IPP
191 IF X(1) < 30 THEN 1670
192 IF X(1) > 40 THEN 1670
193 IF X(2) < .01 THEN 1670
194 IF X(2) > 1 THEN 1670
195 IF X(3) < .0001 THEN 1670
196 IF X(3) > 1 THEN 1670
197 IF X(4) < 15 THEN 1670
198 IF X(4) > 20 THEN 1670
199 IF X(5) < 15 THEN 1670
200 IF X(5) > 20 THEN 1670
202 IF X(6) < .1 THEN 1670
203 IF X(6) > 1 THEN 1670
204 IF X(3) ^ .3333333 * X(4) - X(5) ^
.5 > -1 THEN 1670
226 IF -X(6) - 2 * X(1) * X(2) * X(3) *
X(4) ^ 4 * X(5) ^ -1 * X(6) > -1 THEN 1670
277 IF X(1) ^ -1 * X(3) ^ -1 * X(4) ^
-1 * X(6) + 5 * X(1) ^ -1 * X(2) ^ .5 * X(5) * X(6) > 1 THEN 1670
463 PD1 = -X(1)
466 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1891 IF M < -30.01 THEN 1999
1899 PRINT A(1), A(2), A(3), A(4), A(5),
A(6), M, JJJJ
1999 NEXT JJJJ
This computer program was
run with qb64v1000-win [102]. The
complete output of one run through 32000 is shown below:
30 .31
.01 15.61 19.95
.43 -30
-19130
30 .44
.01 15.64 19.9
.1 -30
22037
30.01 .18
.01 15.8 19.49
.62 -30.01 23855
30.01 .32
.01 15.3 19.75
.19 -30.01 31850
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 Pentium CPU G620@ 2.60GHz, 4.00 GB of
RAM, 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (not
CPU time) for obtaining the output through JJJJ = 32000 was 12 minutes,
counting from "Starting program...".
One can compare the computational results above with those in Wang,
Zhang, and Gao [99, p. 1515, Table 1,
Example 2].
Acknowledgement
I would like to acknowledge
the encouragement of Roberta Clark and Tom Clark.
References
[1] Siby Abraham, Sugata
Sanyal, Mukund Sanglikar (2010), Particle Swarm Optimisation Based Diophantine
Equation Solver, Int. J. of Bio-Inspired Computation, 2 (2), 100-114, 2010.
[2] Siby Abraham, Sugata
Sanyal, Mukund Sangrikar (2013), A Connectionist Network Approach to Find
Numerical Solutions of Diophantine Equations, Int. J. of Engg. Science and
Mgmt., Vol. III, Issue 1, January-June 2013.
[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] 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.
[10] 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.
[11] Hirak Basumatary (1
January 2019). Solve system of equations and inequalities with multiple
solutions?
[12 ] Ahmad Bazzi (January
20, 2022). Multidimensional Newton--Approximate nonlinear equations by sequence
of linear equations--lecture 6.
(Youtube is where I saw this
work.)
[13] Madhulima Bhandari (24
February 2015). How to solve 6 nonlinear coupled equations with 6 unkowns by
MATLAB?
[14] Jerome Bracken, Garth
P. McCormick, Selected Applications of Nonlinear Programming. New York: John
Wiley and Sons, Inc., 1968.
[15] Brian D. Bunday (1984),
Basic Optimisation Methods. London:
Edward Arnold (Publishers) Ltd., 1984.
[16] Richard L. Burden,
Douglas J. Faires, Annette M. Burden, Numerical Analysis, Tenth Edition, 2016,
Cengage Learning.
[17] R. C. Carlson and G. L.
Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol.
14, No. 1 (Jan. – Feb., 1966), pp. 52-58.
[18] Matthew Chan, Yillian
Yin, Brian Amado, Peter Williams (December 21, 2020). Optimization with
absolute values.
https://optimization.cbe.cornell.edu//php?title=Optimization_with_absolute_values#Numerical_
Example
[19] Ta-Cheng Chen (2006).
IAs based approach for reliability redundany allocation problems. Applied
Mathematics and Computation 182 (2006) 1556-1567.
[20] 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.
[21] William Conley (1981).
Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New
York.
[22] H. W. Corley, E. O.
Dwobeng (2020). Relating optimization problems to systems of inequalities and
equalities, American Journal of Operations Research, 2020, 10, 284-298. https://www.scirp.org/journal/ajor.
[23] 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.
[24] George B. Dantzig,
Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr.,
1957), pp. 266-277.
[25] 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.)
[26] 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.
[27] Wassila Drici, Mustapha
Moulai (2019): An exact method for solving multi-objective integer indefinite
quadratic programs, Optimization Methods and Software.
[28] R. J. Duffin, E. L.
Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).
[29] Joseph G. Ecker,
Michael Kupferschmid (1988). Introduction to Operations Research, John Wiley
& Sons, New York (1988).
[30] 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.
[31] 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.
[32] 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.
[33] C. A. Floudas, P. M.
Pardalos, A Collection of Test Problems for Constrained Global Optimization
Algorithms. Springer-Verlag, 1990.
[34] Benjamin Granger, Marta
Yu, Kathleen Zhou (Date Presented: May 25, 2014), Optimization with absolute
values. https://optimization.mccormick.northwestern.edu/index.php/Optimization_with_absolute_values...
[35] Ignacio E. Grossmann.
Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
[36] R. Gupta, R. Malhotra
(1995). Multi-criteria integer linear fractional programming problem,
Optimization, 35:4, 373-389.
[37] Mohammad Babul Hasan,
Sumi Acharjee (2011), Solving LFP by converting it into a single LP,
International Journal of Operations Research, vol. 8, no. 3, pp. 1-14 (2011).
http://www.orstw.org.tw/ijor/vol8no3/1_Vol_8...
[38] Frederick S. Hillier,
Gerald J. Lieberman, Introduction to Operations Research, Ninth Edition, McGraw
Hill, Boston, 2010.
[39] Willi Hock, Klaus
Schittkowski, Test Exalor signomiamples for Nonlinear Programming Codes.
Berlin: Springer-Verlag, 1981.
[40] Xue-Ping Hou, Pei-Ping
Shen, Yong-Qiang Chen, 2014, A global optimization algorithm for signomial
geometric programming problems,
Abstract and Applied
Analysis, vol. 2014, article ID 163263, 12 pages. Hindawi Publishing Corp., http://dx.doi.org/10.1155/2014/163263
[41] Sana Iftekhar, M. J.
Ahsan, Qazi Mazhar Ali (2015). An optimum multivariate stratified sampling
design. Research Journal of Mathematical and Statistical Sciences, vol.
3(1),10-14, January (2015).
[42] R. Israel, A
Karush-Kuhn-Tucker Example
https://personal.math.ubc.ca/~israel/m340/kkk2.pdf
[43] 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.
[44] 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.
[45] 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.
[46] 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.
[47] 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.
[48] 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.
[49] A. H. Land, A. G. Doig,
An Automatic Method of Solving Discrete Programming Problems. Econometrica,
Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[50] 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.
[51] H.-L. Li, H.-C. Lu, Global optimization for generalized geometric
programs with mixed free-sign variables, Operations Research, 57 (2009),
701-713.
[52] Han-Lin Li, Jung-Fa
Tsai (2005). Treating free variables in
generalized geometric global optimization programs. Journal of Global Optimization (2005)
33:1-13.
[53] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of
signomial discrete programming problems with free variables, Optimization and
Engineering (2011) 12: 425-443
[54] 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.
[55] Gia-Shi Liu (2006), A
combination method for reliability-redundancy optimization, Engineering
Optimization, 38:04, 485-499.
[56] 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.
[57] Hao-Chun Lu (2012). An efficient convexification method for
solving generalized geometric problems.
Journal of Industrial and Management Optimization, Volume 8, Number 2,
May 2012, pp. 429-455.
[58] Costas D. Maranas,
Christodoulos A. Floudas, Global Optimization in Generalized Geometric
Programming, pp. 1-42. https://pennstate.pure.elsevier.com/en/publications/global-optimization-in-generalized-geometric-programming
[59] 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.
[60] 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.
[61] 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.
[62] Riley Murray, Venkat
Chandrasekaran, Adam Wierman. Signomial
and polynomial optimization via relative entropy and partial dualization.
[math.OC] 21 July 2019. eprint: arXve:1907.00814v2.
[63] 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.
[64] 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.
[65] 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.
[66] 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.
[67] Max M. J. Opgenoord,
Brian S. Cohn, Warren W. Hobburg (August 31, 2017). Comparison of algorithms for including
equality constraints in signomial programming.
ACDL Technical Report TR-2017-1.
August 31 2017. pp.1-23.
One can get a Google view of
this report.
[68] 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.
[69] 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.
[70] Panos Y. Papalambros,
Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge
University Press, 2000.
[71] O. Perez, I. Amaya, R.
Correa (2013), Numerical solution of certain exponential and nonlinear
diophantine systems of equations by using a discrete particles swarm
optimization algorithm. Applied Mathematics and Computation, Volume 225, 1
December, 2013, pp. 737-746.
[72] 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.
[73] Rajgopal, Geometric
Programming. https://sites.pitt.edu/~jrclass/notes6.pdf
[74] R. V. Rao, P. J. Pawar,
J. P. Davim (2010). Parameter optimization of ultrasonic machining pr5cess
using nontraditional optimization algorihms. Materials and Manufacturing
Processes, 25 (10),1120-1130.
[75] John Rice, Numerical
Methods, Software, and Analysis, Second Edition, 1993, Academic Press.
[76] M. J. Rijckaert, X. M.
Martens, Comparison of generalized geometric programming algorithms, J. of
Optimization, Theory and Applications, 26 (2) 205-242 (1978).
[77] 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.
[78] 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.
[79] 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).
[80] 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.
[81] Vikas Sharma, Kalpana
Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic
fractional integer programming problems, Optimization, 66:11, 1913-1929.
[82] Donald M. Simmons
(1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations
Research, Vol. 17, No. 5 (Sep. – Oct., 1969), pp. 812-826.
[83] 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.
[84] 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
[85] 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.
[86] 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.
[87] Jung-Fa Tsai, Han-Lin
Li, Nian-Ze Hu (2002). Global optimization
for signomial discrete programming problems in engineering design, Engineering
Optimization, 34:6, 613-622.
[88] Jung-Fa Tsai, Ming-Hua
Lin, Yi-Chung Hu (2007). On generalized
geometric programming problems with non-positive variables. European Journal of Operational Research 178
(2007) 10-19.
[89] Jung-Fa Tsai
(2009). Treatng free variables in
generalized geometric programming problems.
Computers and Chemical Enginering 33 (2009) 239-243.
[90] Jung-Fa Tsai, Ming-Hua
Lin (2013). An improved framewpork for
solving NLIPs with signomial terms in the objective or constraints to global
optimality, Computers and Chemical Engineering 53 (2013) 44-54.
[91] Jung-Fa Tsai, Ming-Hua
Lin, Duan-Yi Wen (16 September 2020).
Global optimization for mixed-discrete structural design. Symmetry 2020, 12, 1529
One can get a Google view of
this article. www.mdpi.com/journal/symmetry.
[92] 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
[93] 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.
[94] 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.
[95] 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.
[96] 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.
[97] 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
[98] 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.
[99 ] Y. Wang,
K. Zhang, Y. Gao (2004), Global Optimization of Generalized Geometric
Programming, Computers and Mathematics with Applications 48 (2004) 1505-1516.
[100] Tawan Wasanapradit,
Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving
mixed-integer nonlinear programming problems using improved genetic algorithms.
Korean Journal of Chemical Engineering 28 (1):32-40 January 2011.
[101] Eric W. Weisstein,
"Euler's Sum of Powers Conjecture." https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.
[102] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[103] Wayne L. Winston,
(2004), Operations Research--Applications and Algorithms, Fourtth Edition,
Brooks/Cole--Thomson Learning, Belmont, California 94002.
[104] Helen Wu, (2015),
Geometric Programming
https://optimization.mccormick.northwestern.edu/index.php/Geometric_Programming
[105] G. Xu, (2014). Global
optimization of signomial geometric programming problems, European J. of
Operational Research 233 (2014) 500-510.
[106] James Yan. Signomial programs with equality constraints:
numerical solution and applications. Ph.
D. thesis. University of British Columbia, 1976.
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