February 25, 2018 by 123writereadmath

Solving a Fuzzy Multicriteria Programming Problem Involving Nonlinear Membership Functions

Jsun Yui Wong

Hannan’s 3-objective problem [13, Example 1, pp. 243-245] has been transformed by Yang, Ignizio, and Kim [47, p. 50] to the following problem:

Maximize lambda

subject to

lambda<= 1 – (6 – (3 * X(1) + X(2) + X(3))) / (2),

lambda<= 1 – (20 / 3 – (3 * X(1) + X(2) + X(3))) / (10 / 3),

lambda <= 1 – (7 – (3 * X(1) + X(2) + X(3))) / (5),

lambda < = 1 – (7 – (X(1) – X(2) + 2 * X(3))) / (5),
lambda <= 1 – (8 – (X(1) – X(2) + 2 * X(3))) / (20 / 3),

lambda <= 1 – (9 / 2 – (X(1) + 2 * X(2))) / (5 / 2),

lambda <= 1 – (5 – (X(1) + 2 * X(2))) / (5),

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

X(1) + 3 * X(2) + 2 * X(3)<=8,

X(3) <=5,

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

The computer program listed below seeks to solve the formulation above–X(5) through X(13) below are slack variables.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO -31999. STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 4
22 A(J44) = RND * 2

31 NEXT J44

128 FOR I = 1 TO 50000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 4)

183 r = (1 – RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP

223 FOR j41 = 1 TO 3

225 IF X(j41) < 0 THEN 1670

235 NEXT j41

239 IF X(3) > 5 THEN 1670

309 X(5) = -X(4) + 1 – (6 – (3 * X(1) + X(2) + X(3))) / (2)

311 X(6) = -X(4) + 1 – (20 / 3 – (3 * X(1) + X(2) + X(3))) / (10 / 3)

313 X(7) = -X(4) + 1 – (7 – (3 * X(1) + X(2) + X(3))) / (5)

315 X(8) = -X(4) + 1 – (7 – (X(1) – X(2) + 2 * X(3))) / (5)
317 X(9) = -X(4) + 1 – (8 – (X(1) – X(2) + 2 * X(3))) / (20 / 3)

319 X(10) = -X(4) + 1 – (9 / 2 – (X(1) + 2 * X(2))) / (5 / 2)

321 X(11) = -X(4) + 1 – (5 – (X(1) + 2 * X(2))) / (5)

323 X(12) = 10 – 4 * X(1) – 2 * X(2) – 3 * X(3)

325 X(13) = 8 – X(1) – 3 * X(2) – 2 * X(3)

337 FOR J44 = 5 TO 13

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

443 POBA = X(4) + 1000000 * (X(10) + X(11) + X(12) + X(13) + X(5) + X(6) + X(7) + X(8) + X(9))
466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 13

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < .2631 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), A(8), A(9), A(10)
1905 PRINT A(11), A(12), A(13), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [46]. The complete output through JJJJ = -31999.82000000003 is shown below:

.5000473359910486 1.078919963472388 1.947323576334272
.263154905037441 0
0 0 0 0 0
0 0 0 .263154905037441
-31999.82000000003

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 [46], the wall-clock time for obtaining the output through
JJJJ= -31999.82000000003 was 70 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above to those on page 51 of Yang, Ignizio, and Kim [47].

Acknowledgment

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

References

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[11] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[18] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

[19] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[20] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[21] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[22] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[23] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[24] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[25] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[26] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[27] 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.
[28] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[29] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[30] 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.
[31] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[32] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[33] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[34] 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.
[35] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[36] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[37] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[38] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[39] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[40] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[41] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[42] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[43] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[44] 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) pp. 10-19.
[45] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[46] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[47] Taeyong Yang, James P. Ignizio, Hyun-Joon Kim (1991), Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.

February 17, 2018 by 123writereadmath

Solving a Multiple Objective Programming Problem Using a Problem Obtained from the Original Problem

Jsun Yui Wong

The following formulation is from Jain and Lachhwani [15, pp. 46-47]:

Maximize {Z1(X), Z2(X)}

where Z1(X)= 2 * X(1) + (2 * X(1)+6*X(2)) / (X(1)+ X(2) + 1),

Z2(X)= 2 * X(2) + ( X(1)+X(2)) / (X(1)- X(2) + 1),

subject to

X(1) +X(2) + X(3)=4,

3*X(1) + X(2) – X(4)=6,

X(1) -X(2+0* X(3))+0* X(4)=0,

where X(1) through X(4) >=0.

The X(1) above, for example, is not the same as the X(1) below– see Jain and Lachhwani [15, pp. 46-47].
The X(4) beow can be better denoted by lambda–see Jain and Lachhwani [15, p. 47].

In [15] Jain and Lachhwani show their procedure for transforming the problem above to the following problem [15, p. 47]:

Maximize X(4)

subject to

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

– X(2) + 4 * X(3)=6,

6 * X(4)<= 2 * X(3) + (8 * X(3)) / (2 * X(3) + 1),

6 * X(4)<= 4 * X(3),

where X(1) through X(4) >=0.

The following computer program uses the second formulation shown above.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 31999 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37

71 FOR J40 = 1 TO 4

75 A(J40) = RND

77 NEXT J40
128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 4)

183 r = (1 – RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP

223 FOR j41 = 1 TO 4

225 IF X(j41) < 0 THEN 1670

235 NEXT j41

261 X(1) = 4 – 2 * X(3)
264 X(2) = -6 + 4 * X(3)

171 IF X(1) < 0 THEN GOTO 1670

173 IF X(2) < 0 THEN GOTO 1670

318 X(5) = 0 + 2 * X(3) + (8 * X(3)) / (2 * X(3) + 1) – 6 * X(4)
319 X(6) = 0 + 4 * X(3) – 6 * X(4)

337 FOR J44 = 5 TO 6

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

433 POBA = X(4) + 1000000 * (X(5) + X(6))

466 P = POBA

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 < 1.2 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [43]. The complete output through JJJJ = -31999.78000000004 is shown below:

1.213658951115804D-08 1.999999975726821 1.999999993931705
1.199999997653591 0 0 1.199999997653591
-31999.96000000001

1.952032668839365D-08 1.999999960959347 1.999999990239837
1.19999999622607 0 0 1.19999999622607
-31999.93000000001

2.275460575518196D-08 1.99999995449079 1.999999988622697
1.199999995600776 0 0 1.199999995600776
-31999.80000000003

7.748838015686488D-09 1.999999984502324 1.999999996125581
1.19999999850189 0 0 1.19999999850189
-31999.78000000004

This solution (7.748838015686488D-09 1.999999984502324 1.999999996125581) at JJJJ= -31999.78000000004 is translated to (2 2 0 2) for the original problem–see Jain and Lachhwani [15, p. 48].

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 [43], the wall-clock time for obtaining the output through
JJJJ= -31999.78000000004 was 15 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above to those on page 48 of Jain and Lachhwani [15].

Acknowledgment

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

References

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[13] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[14] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[15] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

[16] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[17] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[18] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[19] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[20] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[21] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[22] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[23] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[24] 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.
[25] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[26] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[27] 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.
[28] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[29] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[30] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[31] 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.
[32] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[33] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[34] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[35] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[36] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[37] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[38] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[39] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[40] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[41] 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) pp. 10-19.
[42] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[43] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[44] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.

February 8, 2018 by 123writereadmath

Solving a Multiple Objective Integer Nonlinear Programming Problem Using the Traditional Utility Function Approach and the Solver Here

Jsun Yui Wong

The computer program listed below seeks to solve the following problem:

Maximize ((X(1) + 2 * X(3)) * X(4) / (X(1) + X(2) + 2)) + ((4 * X(1) + 2 * X(2)) * X(5) / (X(2) + X(3) + 2)) + ((2 * X(1) + X(2)) * X(6) / (X(1) + X(3) + 1)), which is the sum of three objectives–see Chang [6, p. 443], Lachhwani [18, p. 321], and MacCrimmon [28, pp.18-44],

subject to

X(1) + X(2)<=1000,

1.5 * X(1) + X(2) + X(3)>=4,

0<= X(1), X(2), X(3)<=1000,

0<= X(4), X(5), X(6)<=1,

X(1), X(2), and X(3) are general integer variables,

X(4), X(5), and X(6) are 0-1 variables.

X(7) and X(8) below are two added slack variables.

The problem above is based on Example 1 of Chang [6, p. 443], but these two are not the same.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 3
55 A(J40) = RND * 4

56 NEXT J40

71 FOR J40 = 4 TO 6
75 A(J40) = RND

77 NEXT J40

128 FOR I = 1 TO 50000

129 FOR KKQQ = 1 TO 6

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 6)

183 r = (1 – RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP

223 FOR j41 = 1 TO 6

225 X(j41) = INT(X(j41))

235 NEXT j41

236 FOR J47 = 1 TO 3

237 IF X(J47) < 0 THEN 1670
238 IF X(J47) > 1000 THEN 1670

239 NEXT J47

246 FOR J47 = 4 TO 6

247 IF X(J47) < 0 THEN 1670
248 IF X(J47) > 1 THEN 1670

249 NEXT J47

318 X(7) = 1000 – X(1) – X(2)

320 X(8) = -4 + 1.5 * X(1) + X(2) + X(3)

337 FOR J44 = 7 TO 8

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

349 POBA = ((X(1) + 2 * X(3)) * X(4) / (X(1) + X(2) + 2)) + ((4 * X(1) + 2 * X(2)) * X(5) / (X(2) + X(3) + 2)) + ((2 * X(1) + X(2)) * X(6) / (X(1) + X(3) + 1)) + 1000000 * (X(7) + X(8))

466 P = POBA

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 < 10 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [42]. The complete output through JJJJ = -31998.1200000003 is shown below:

1000 0 0 0 1
0 0 0 2000 -31999.74000000004

0 0 1000 1 0
0 0 0 1000 -31999.56000000007

0 1000 0 0 0
1 0 0 1000 -31998.1300000003

1000 0 0 0 1
0 0 0 2000 -31998.1200000003

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 [42], the wall-clock time for obtaining the output through
JJJJ= -31998.1200000003 was 30 seconds, total, including the time for “Creating .EXE file.”

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[11] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[14] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[15] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[16] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[17] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[18] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[19] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[20] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[21] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[22] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[23] 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.
[24] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[25] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[26] 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.
[27] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[28] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[29] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[30] 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.
[31] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[32] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[33] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[34] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[35] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[36] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[37] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[38] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[39] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[40] 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) pp. 10-19.
[41] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[42] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[43] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.

February 7, 2018 by 123writereadmath

Solving a Multi Objective Programming Problem Using the Traditional Utility Approach

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from page 321 of Lachhwani [17]:

Maximize 6 + 9 * X(2) – 2 * X(1) ^ 2 – 4 * X(1) * X(2), which is the sum of three objectives–see Lachhwani [17, p. 321] and MacCrimmon [27, pp.18-44],

subject to

X(1) + X(2)<=2,

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

X(3) below is an added slack variable.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 2
55 A(J40) = RND

56 NEXT J40

128 FOR I = 1 TO 20000

129 FOR KKQQ = 1 TO 2

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

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

181 j = 1 + FIX(RND * 8)

183 r = (1 – RND * 2) * A(j)

187 X(j) = A(j) + (RND ^ (RND * 10)) * r

222 NEXT IPP
314 FOR J47 = 1 TO 2
316 IF X(J47) < 0 THEN 1670

317 NEXT J47
327 X(3) = 2 – X(1) – X(2)
337 FOR J44 = 3 TO 3

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44
455 POBA = 6 + 9 * X(2) – 2 * X(1) ^ 2 – 4 * X(1) * X(2) + 1000000 * (X(3))
466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 3

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

1457 M = P

1559 GOTO 128

1670 NEXT I
1889 IF M < -.000001 THEN 1999

1900 PRINT A(1), A(2), A(3)

1903 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [41]. The complete output through JJJJ =-31999.95000000001 is shown below:

4.3846605288648D-18 2 0
24 -32000

1.922376176184634D-17 2 0
24 -31999.99

2.052857656225404D-17 1.999999999999982 0
23.99999999999984 -31999.98

4.128598381258941D-20 2 0
24 -31999.97000000001

3.492463152665762D-18 2 0
24 -31999.96000000001

2.134532590413878D-17 2 0
24 -31999.95000000001

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 [41], the wall-clock time for obtaining the output through
JJJJ= -31999.95000000001 was 3 seconds, not including the time for “Creating .EXE file.” One can compare the computational results above with those on page 321 of Lachhwani [17].

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[12] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[13] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[14] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[15] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

[16] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[17] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[18] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[27] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University os South Carolina Press, Columbia.
[28] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[29] 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.
[30] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[31] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[32] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[33] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[34] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[35] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[36] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[37] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[38] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[39] 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) pp. 10-19.
[40] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[41] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[42] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.

February 4, 2018 by 123writereadmath

Solving Another Multi Objective Quadratic Fractional Problem with Lachhwani’s Fuzzy Goal Programming Approach and the Computer Program Here

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Lachhwani [17, pp. 320-321, Example 2]:

Minimize X(3) + X(5) + X(7), which are three deviation variables,

subject to

6 – 6 * X(1) + 2 * X(1) ^ 2 – 2 * X(1) * X(2) + 2 * X(2) ^ 2 – 14 + 13.50 * X(3) – 13.50*X(4) =0,

4 * X(1) + 6 * X(2) – 2 * X(1) ^ 2 – 2 * X(1) * X(2) – 4.6667 + 13.50 * X(5) -13.50* X(6) =0,

2 * X(1) + 3 * X(2) – 2 * X(1) ^ 2 – 6 + 13.50 * X(7) – 13.50*X(8) = 0,

X(1) + X(2)<=2,

X(1), X(2)>=0,

X(3) through X(8) are deviation variables.

X(9) below is an added slack variable.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 8
55 A(J40) = RND

56 NEXT J40
66 A(1) = RND

68 A(2) = RND

128 FOR I = 1 TO 200000

129 FOR KKQQ = 1 TO 8

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

132 REM IF RND < .5 THEN GOTO 133 ELSE GOTO 249

133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 8)

183 r = (1 – RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r

222 NEXT IPP

257 X(4) = (6 – 6 * X(1) + 2 * X(1) ^ 2 – 2 * X(1) * X(2) + 2 * X(2) ^ 2 – 14 + 13.5 * X(3)) / 13.5

258 X(6) = (4 * X(1) + 6 * X(2) – 2 * X(1) ^ 2 – 2 * X(1) * X(2) – 4.6667 + 13.5 * X(5)) / 13.5

261 X(8) = (2 * X(1) + 3 * X(2) – 2 * X(1) ^ 2 – 6 + 13.5 * X(7)) / 13.5

267 X(9) = 2 – X(1) – X(2)
314 FOR J47 = 1 TO 8

315 REM IF X(J47) > 1 THEN 1670
316 IF X(J47) < 0 THEN 1670
317 NEXT J47

337 FOR J44 = 9 TO 9

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

453 POBA = -X(3) – X(5) – X(7) + 1000000 * (X(9))

466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 9

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

1457 M = P

1559 GOTO 128

1670 NEXT I
1889 IF M < -.000001 THEN 1999

1900 PRINT A(1), A(2), A(3)

1901 PRINT A(4), A(5), A(6)

1904 PRINT A(7), A(8), A(9)

1903 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ =-31999.95000000001 is shown below:

3.575691903814191D-11 1.99999999996424 4.767762271661601D-11
3.350847388814348D-27 7.21331790695476D-274 .5432074073915142
2.649309655400747D-12 2.498176401482126D-27 0
-5.032693237201676D-11 -32000

9.183328289590226D-10 1.999999999081667 1.224443720733797D-09
1.991360733923955D-25 4.95571589969331D-129 .5432074069992595
6.802463505036006D-11 1.589259047266234D-25 0
-1.292468355784157D-09 -31999.95000000001

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.95000000001 was 50 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above with those on page 321 of Lachhwani [17].

Acknowledgment

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

References

[13] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[14] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[16] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[17] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[18] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[19] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[20] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[21] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[22] 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.
[23] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[24] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[25] 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.
[26] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[27] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[28] 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.
[29] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[30] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[31] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[32] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[33] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[34] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[35] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[36] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[37] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[38] 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) pp. 10-19.
[39] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.

February 2, 2018 by 123writereadmath

Solving a Multi Objective Quadratic Fractional Problem with Lachhwani’s Fuzzy Goal Programming Approach and with the Solver Here

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Lachhwani [17, pp. 54-55, Model III]:

Minimize X(3) + X(4), which are two deviational variables,

subject to

((2 * X(1) + 20 * X(2) + 12) * (X(1) + 10 * X(2) + 17) * X(5)) + .43653 * X(3) – 1.67289>=0,

((2 * X(1) + 20 * X(2) + 12) * (3 * X(1) + 30 * X(2) + 51) * X(6)) + .43653 * X(4) – 2.50934>=0,

((-2 * X(1) – 5 * X(2) + 15) * (2 * X(1) + 5 * X(2) + 11))*X(5)=1,

((-4 * X(1) – 10 * X(2) + 30) * (2 * X(1) + 5 * X(2) + 11))*X(6)=1,

X(1) + 15 * X(2)<=2,
3 * X(1) + 20 * X(2)<=4,

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

X(5), X(6)>0.

X(7) through X(10) below are added slack variables.

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 6
55 A(J40) = RND

56 NEXT J40

128 FOR I = 1 TO 100000

129 FOR KKQQ = 1 TO 6

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

181 j = 1 + FIX(RND * 6)

183 r = (1 – RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP

231 X(5) = 1 / ((-2 * X(1) – 5 * X(2) + 15) * (2 * X(1) + 5 * X(2) + 11))

234 X(6) = 1 / ((-4 * X(1) – 10 * X(2) + 30) * (2 * X(1) + 5 * X(2) + 11))
244 FOR J47 = 1 TO 4

246 IF X(J47) < 0 THEN 1670
247 NEXT J47

256 X(7) = ((2 * X(1) + 20 * X(2) + 12) * (X(1) + 10 * X(2) + 17) * X(5)) + .43653 * X(3) – 1.67289

258 X(8) = ((2 * X(1) + 20 * X(2) + 12) * (3 * X(1) + 30 * X(2) + 51) * X(6)) + .43653 * X(4) – 2.50934

267 X(9) = 2 – X(1) – 15 * X(2)
268 X(10) = 4 – 3 * X(1) – 20 * X(2)

337 FOR J44 = 7 TO 10

338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

451 POBA = -X(3) – X(4) + 1000000 * (X(7) + X(8) + X(9) + X(10))

466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 10

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

1457 M = P

1559 GOTO 128

1670 NEXT I
1889 IF M < -.000001 THEN 1999

1900 PRINT A(1), A(2), A(3)

1901 PRINT A(4), A(5), A(6)
1902 PRINT A(7), A(8), A(9)

1903 PRINT A(10), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [39]. The complete output through JJJJ =-31999.97000000001 is shown below:

.8000135435080251 7.999720673683235D-02 0
0 5.917159763319737D-03 2.958579881659818D-03
0 0 0
0 0 -31999.98

.8000020249335132 7.99907841871481D-02 0
0 5.917159763313621D-03 2.95857988165681D-03
0 0 0
0 0 -31999.97000000001

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 [39], the wall-clock time for obtaining the output through JJJJ= -31999.97000000001 was 18 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above with those on page 55 of Lachhwani [17].

Acknowledgment

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

References

[13] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

[14] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[16] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[17] K. Lachhwani (2014). FGP approach to multi objectve quadratic fraction programming problem. Indian Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57.
Available online at http://ijim.srbiau.ac.ir/

[18] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[19] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[20] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[21] 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.
[22] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
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https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[27] 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.
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[38] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[39] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[40] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.

computerprogramsandresults

Nonlinear Integer Programming

Month: February 2018

Solving a Fuzzy Multicriteria Programming Problem Involving Nonlinear Membership Functions

Solving a Multiple Objective Programming Problem Using a Problem Obtained from the Original Problem

Solving a Multiple Objective Integer Nonlinear Programming Problem Using the Traditional Utility Function Approach and the Solver Here

Solving a Multi Objective Programming Problem Using the Traditional Utility Approach

Solving Another Multi Objective Quadratic Fractional Problem with Lachhwani’s Fuzzy Goal Programming Approach and the Computer Program Here

Solving a Multi Objective Quadratic Fractional Problem with Lachhwani’s Fuzzy Goal Programming Approach and with the Solver Here