Straight Solving Linear Fractional Programming Problems: An Illustration

 

Straight Solving Linear Fractional Programming Problems: An Illustration

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below seeks to solve the following problem from Hasan and Acharjee [37, p. 12, Numerical Example 5]:   

Maximize      (X(1) + 2 * X(2) + 3.5 * X(3) + X(4) + 1) / (2 * X(1) + 2 * X(2) + 3.5 * X(3) + 3 * X(4) + 4)

  

subject to

         2 * X(1) + X(2) + 3 * X(3) + 3 * X(4) <= 10,

         X(1) + 2 * X(2) + X(3) + X(4) <= 14,

     

         X(1),  X(2), X(3),  X(4) >= 0.

  

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), 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)

85 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3D+30

    111 FOR J44 = 1 TO 4

        112 A(J44) = FIX(RND * 21)

    113 NEXT J44

    128 FOR I = 1 TO 150000

        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

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

            140 B = 1 + FIX(RND * 4)

            144 IF RND < .5 THEN 160 ELSE GOTO 167

            160 R = (1 - RND * 2) * A(B)

            164 X(B) = A(B) + (RND ^ (RND * 15)) * R

            165 GOTO 168

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 3) ELSE X(B) = A(B) + FIX(RND * 3)

        168 NEXT IPP

        171 IF X(1) < 0## THEN 1670

        175 IF X(2) < 0## THEN 1670

        177 IF X(3) < 0## THEN 1670

        179 IF X(4) < 0## THEN 1670

        404 IF 2 * X(1) + X(2) + 3 * X(3) + 3 * X(4) > 10## THEN 1670

        406 IF X(1) + 2 * X(2) + X(3) + X(4) > 14## THEN 1670

     

        409 PD1 = (X(1) + 2 * X(2) + 3.5 * X(3) + X(4) + 1) / (2 * X(1) + 2 * X(2) + 3.5 * X(3) + 3 * X(4) + 4)

        466 P = PD1

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1779 IF M < -9999 THEN 1999

    1904 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102].  Its output of one run through JJJJ=-31899 is shown below:

0      6.40067550592987        1.198647516960092  

0      .85711984201531         -31982

1.391493333437622D-16      6.398962636323966       1.200345787777984  

2.065218353606715D-17      .857136976151587        -31906

0      6.399999127939391      1.200000290686772  

3.125690125099123D-19      .8571428521992004      -31899

One distinct solution is shown above.

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 (3.89 GB usable), 64-bit Operating System, and QB64v1000-win [102], the wall-clock time (not CPU time) for obtaining the output through  JJJJ  = -31899 was 17 seconds, counting from "Starting program...".   

   

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?

https://www.mathworks.com/matlabcentral/answers/437815-solve-system-of-equations-and-inequalities-with-multiple-solutions?_tid=prof_contriblnk 

[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.)

http://bazziahmad.com/

[13] Madhulima Bhandari (24 February 2015).   How to solve 6 nonlinear coupled equations with 6 unkowns by MATLAB?

https://mathworks.com/matlabcentral/answers/180104-how-to-solve-6-nonlinear-coupled-equations-with-6-unkowns-by-matlab

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

[14]  https://www.brainkart.com/article/Inequality-Constraints-Karush-Kuhn-Tucker-(KKT)-Conditions_11264/

[15] 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.

[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 Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.

[40] 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).

[41] R. Israel,  A Karush-Kuhn-Tucker Example

https://personal.math.ubc.ca/~israel/m340/kkk2.pdf

[42] 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

[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, 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.

[49] 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.

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

[51] 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.

[52] 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.

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

[54] 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.

[55] 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.

[56] 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

[57] 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.

[58] 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.

[59] 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.

[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] 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.)

[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] 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.

[68] 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/

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

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

[72]  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.

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

[74] 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.

[75] 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.

[76]  John Rice, Numerical Methods, Software, and Analysis, Second Edition, 1993, Academic Press.

[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] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. ht3ps://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf

[84] 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.

[85] 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

[86] 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.

[87] 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.

[88] 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

[89] 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.

[90] 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.

[91] 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.

[92] 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.

[93] 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.

[94] 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

[95] 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.

[96] 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.

[97]  Eric W. Weisstein, "Diophantine Equation--8th Powers."  https://mathworld.wolfram.com/DiophantineEquation8thPowers.html.

[98]  Eric W. Weisstein, "Euler's Sum of Powers Conjecture."  https://mathworld.wolfram.com/EulersSumofPowersConjecture.html.

[99]  Eric W. Weisstein, "Diophantine Equation--5th Powers."  https://mathworld.wolfram.com/DiophantineEquation5thPowers.html.

[100]  Eric W. Weisstein, "Diophantine Equation--10th Powers."  https://mathworld.wolfram.com/DiophantineEquation10thPowers.html.

[101]  Eric W. Weisstein, "Diophantine Equation--9th Powers."  https://mathworld.wolfram.com/DiophantineEquation9thPowers.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] Jsun Yui Wong (07/04/2016).  A Computer Program with Additional Domino Effect Solving a Nonlinear Diophantine System of 10 Integer Unknowns and 9 Equations, Second Edition.  Retrieved from https://myblogsubstance.typepad.com/substance/2016/07.

[105] 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.