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Filled function method to optimize supply chain transportation costs
1. | School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471000, China |
2. | School of Information Engineering, Henan University of Science and Technology, Luoyang 471000, China |
The transportation-based supply chain model can be formulated as the constrained nonlinear programming problems. When solving such problems, the classic optimization algorithms are often limited to local minimums, causing the difficulty to find the global optimal solution. Aiming at this problem, a filled function method with a single parameter is given to cross the local minimum. Based on the characteristics of the filled function, a new filled function algorithm that can obtain the global optimal solution is designed. Numerical experiments verify the feasibility and effectiveness of the algorithm. Finally, the filled function algorithm is applied to the solution of supply chain problems, and the numerical results show that the algorithm can also address decision-making problems of supply chain transportation effectively.
References:
[1] |
R. P. Ge,
A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204.
doi: 10.1007/BF01585737. |
[2] |
R. P. Ge and Y. F. Qin,
A class of filled function for finding global minimizer of a function of several variables, Journal of Optimization Theory and Applicaions, 54 (1987), 241-252.
doi: 10.1007/BF00939433. |
[3] |
A. V. Levy and A. Montalvo,
The tunneling algorithm for the global minimization of functions, SIAM Journal on Scientific and Statistical Computing, 6 (1985), 15-29.
doi: 10.1137/0906002. |
[4] |
J. Y. Li, B. S. Han and Y. J. Yang,
A novel one parameter filled function, Comm. on Appl. Math. and Comput, 24 (2010), 17-24.
|
[5] |
J. R. Li, Y. L. Shang and P. Han,
New Tunnel-Filled Function Method for Discrete Global Optimization, Journal of the Operations Research Society of China, 5 (2017), 291-300.
doi: 10.1007/s40305-017-0160-8. |
[6] |
H. W. Lin, Y. L. Gao, X. Wang and S. Su,
A filled function which has the same local minimizer of the objective function, Optimization Letters, 13 (2019), 761-776.
doi: 10.1007/s11590-018-1275-5. |
[7] |
Y. L. Shang, Research on Filled Function Method in Nonlinear Global Optimization, Ph.D thesis, Shanghai University in Shanghai of China, 2005. |
[8] |
Y. L. Shang and L. S. Zhang,
Finding discrete global minima with a filled function for integer programming, European Journal of Operational Research, 189 (2008), 31-40.
doi: 10.1016/j.ejor.2007.05.028. |
[9] |
L. Y. Shu and Q. P. Yan,
Study of a non-linear optimal model on the manufacturer core supply chain, Systems Engineering-Theory and Practice, 2 (2006), 36-41.
|
[10] |
W. X. Wang, Y. L. Shang and D. Wang,
Filled function method for solving non-smooth box constrained global optimization problems, Operational Research Transactions, 23 (2019), 28-34.
|
[11] |
W. X. Wang, Y. L. Shang and L. S. Zhang,
A filled function method with one parameter for constrained global optimization, Chinese Journal of Engineering Mathematics, 25 (2008), 795-803.
|
[12] |
Y. Wang, W. Fang and T. Wu,
A cut-peak function method for global optimization, Journal of Computational and Applied Mathematics, 230 (2009), 135-142.
doi: 10.1016/j.cam.2008.10.069. |
[13] |
Y. J. Yang, M. L. He and Y. L. Gao,
Discrete Global Optimization Problems with a Modified Discrete Filled Function, Journal of the Operations Research Society of China, 3 (2015), 297-315.
doi: 10.1007/s40305-015-0085-z. |
[14] |
Y. J. Yang and Y. M. Liang,
A new discrete filled function algorithm for discrete global optimization, Journal of Computational and Applied Mathematics, 202 (2007), 280-291.
doi: 10.1016/j.cam.2006.02.032. |
[15] |
Y. J. Yang and Y. L. Shang,
A new filled function method for unconstrained global optimization, Applied Mathematics and Computation, 173 (2006), 501-512.
doi: 10.1016/j.amc.2005.04.046. |
[16] |
Y. J. Yang, Z. Y. Wu and F. S. Bai,
A filled function method for constrained nonlinear integer programming, Journal of Industrial and Management Optimization, 4 (2008), 353-362.
doi: 10.3934/jimo.2008.4.353. |
[17] |
L. Yuan, Z. Wan and Q. Tang,
A criterion for an approximation global optimal solution based on the filled functions, Journal of Industrial and Management Optimization, 12 (2016), 375-387.
doi: 10.3934/jimo.2016.12.375. |
[18] |
L. Yuan, Z. Wan, J. Zhang and B. Sun,
A filled function method for solving nonlinear complementarity problem, Journal of Industrial and Management Optimization, 5 (2009), 911-928.
doi: 10.3934/jimo.2009.5.911. |
[19] |
Y. Zhang, L. S. Zhang and Y. T. Xu,
New filled functions for non-smooth global optimization, Applied Mathematical Modelling, 33 (2009), 3114-3129.
doi: 10.1016/j.apm.2008.10.015. |
show all references
References:
[1] |
R. P. Ge,
A filled function method for finding a global minimizer of a function of several variables, Mathematical Programming, 46 (1990), 191-204.
doi: 10.1007/BF01585737. |
[2] |
R. P. Ge and Y. F. Qin,
A class of filled function for finding global minimizer of a function of several variables, Journal of Optimization Theory and Applicaions, 54 (1987), 241-252.
doi: 10.1007/BF00939433. |
[3] |
A. V. Levy and A. Montalvo,
The tunneling algorithm for the global minimization of functions, SIAM Journal on Scientific and Statistical Computing, 6 (1985), 15-29.
doi: 10.1137/0906002. |
[4] |
J. Y. Li, B. S. Han and Y. J. Yang,
A novel one parameter filled function, Comm. on Appl. Math. and Comput, 24 (2010), 17-24.
|
[5] |
J. R. Li, Y. L. Shang and P. Han,
New Tunnel-Filled Function Method for Discrete Global Optimization, Journal of the Operations Research Society of China, 5 (2017), 291-300.
doi: 10.1007/s40305-017-0160-8. |
[6] |
H. W. Lin, Y. L. Gao, X. Wang and S. Su,
A filled function which has the same local minimizer of the objective function, Optimization Letters, 13 (2019), 761-776.
doi: 10.1007/s11590-018-1275-5. |
[7] |
Y. L. Shang, Research on Filled Function Method in Nonlinear Global Optimization, Ph.D thesis, Shanghai University in Shanghai of China, 2005. |
[8] |
Y. L. Shang and L. S. Zhang,
Finding discrete global minima with a filled function for integer programming, European Journal of Operational Research, 189 (2008), 31-40.
doi: 10.1016/j.ejor.2007.05.028. |
[9] |
L. Y. Shu and Q. P. Yan,
Study of a non-linear optimal model on the manufacturer core supply chain, Systems Engineering-Theory and Practice, 2 (2006), 36-41.
|
[10] |
W. X. Wang, Y. L. Shang and D. Wang,
Filled function method for solving non-smooth box constrained global optimization problems, Operational Research Transactions, 23 (2019), 28-34.
|
[11] |
W. X. Wang, Y. L. Shang and L. S. Zhang,
A filled function method with one parameter for constrained global optimization, Chinese Journal of Engineering Mathematics, 25 (2008), 795-803.
|
[12] |
Y. Wang, W. Fang and T. Wu,
A cut-peak function method for global optimization, Journal of Computational and Applied Mathematics, 230 (2009), 135-142.
doi: 10.1016/j.cam.2008.10.069. |
[13] |
Y. J. Yang, M. L. He and Y. L. Gao,
Discrete Global Optimization Problems with a Modified Discrete Filled Function, Journal of the Operations Research Society of China, 3 (2015), 297-315.
doi: 10.1007/s40305-015-0085-z. |
[14] |
Y. J. Yang and Y. M. Liang,
A new discrete filled function algorithm for discrete global optimization, Journal of Computational and Applied Mathematics, 202 (2007), 280-291.
doi: 10.1016/j.cam.2006.02.032. |
[15] |
Y. J. Yang and Y. L. Shang,
A new filled function method for unconstrained global optimization, Applied Mathematics and Computation, 173 (2006), 501-512.
doi: 10.1016/j.amc.2005.04.046. |
[16] |
Y. J. Yang, Z. Y. Wu and F. S. Bai,
A filled function method for constrained nonlinear integer programming, Journal of Industrial and Management Optimization, 4 (2008), 353-362.
doi: 10.3934/jimo.2008.4.353. |
[17] |
L. Yuan, Z. Wan and Q. Tang,
A criterion for an approximation global optimal solution based on the filled functions, Journal of Industrial and Management Optimization, 12 (2016), 375-387.
doi: 10.3934/jimo.2016.12.375. |
[18] |
L. Yuan, Z. Wan, J. Zhang and B. Sun,
A filled function method for solving nonlinear complementarity problem, Journal of Industrial and Management Optimization, 5 (2009), 911-928.
doi: 10.3934/jimo.2009.5.911. |
[19] |
Y. Zhang, L. S. Zhang and Y. T. Xu,
New filled functions for non-smooth global optimization, Applied Mathematical Modelling, 33 (2009), 3114-3129.
doi: 10.1016/j.apm.2008.10.015. |
The proposed Algorithm | Algorithm in [10] | |
Problem 1 | -1.0316 | -1.0316 |
Problem 2 | 0 | 0 |
Problem 3 | 1.513e-08 | 0 |
Problem 4, n=10 | 4.4277e-43 | 1.3790e-14 |
Problem 4, n=20 | 3.2490e-44 | 3.0992e-14 |
Problem 4, n=50 | 1.8410e-43 | 9.85.1e-13 |
The proposed Algorithm | Algorithm in [10] | |
Problem 1 | -1.0316 | -1.0316 |
Problem 2 | 0 | 0 |
Problem 3 | 1.513e-08 | 0 |
Problem 4, n=10 | 4.4277e-43 | 1.3790e-14 |
Problem 4, n=20 | 3.2490e-44 | 3.0992e-14 |
Problem 4, n=50 | 1.8410e-43 | 9.85.1e-13 |
The proposed Algorithm | Algorithm in [10] | |||
CPU run time(s) | Total times(times) | CPU run time(s) | Total times(times) | |
Problem 1 | 19.1964 | 1986 | 26.3325 | 2453 |
Problem 2 | 15.1005 | 1537 | 18.8756 | 1421 |
Problem 3 | 10.8827 | 961 | 14.3194 | 1227 |
Problem 4, n=10 | 16.0294 | 1920 | 70.5483 | 8895 |
Problem 4, n=20 | 24.6212 | 4696 | 91.3288 | 18242 |
Problem 4, n=50 | 34.9223 | 6728 | 195.3385 | 43232 |
The proposed Algorithm | Algorithm in [10] | |||
CPU run time(s) | Total times(times) | CPU run time(s) | Total times(times) | |
Problem 1 | 19.1964 | 1986 | 26.3325 | 2453 |
Problem 2 | 15.1005 | 1537 | 18.8756 | 1421 |
Problem 3 | 10.8827 | 961 | 14.3194 | 1227 |
Problem 4, n=10 | 16.0294 | 1920 | 70.5483 | 8895 |
Problem 4, n=20 | 24.6212 | 4696 | 91.3288 | 18242 |
Problem 4, n=50 | 34.9223 | 6728 | 195.3385 | 43232 |
Transporter 1 | Transporter 2 | ||||
Mode of generalized transport | unit cost($/t) | maximum amount(t) | unit cost($/t) | maximum amount(t) | |
Seller 1 | 220 | 200 | |||
Transporter 1 | Seller 2 | 250 | 1000 | 220 | 1500 |
Seller 3 | 210 | 210 | |||
Seller 1 | 180 | 200 | |||
Transporter 2 | Seller 2 | 200 | 1200 | 210 | 1000 |
Seller 3 | 210 | 220 |
Transporter 1 | Transporter 2 | ||||
Mode of generalized transport | unit cost($/t) | maximum amount(t) | unit cost($/t) | maximum amount(t) | |
Seller 1 | 220 | 200 | |||
Transporter 1 | Seller 2 | 250 | 1000 | 220 | 1500 |
Seller 3 | 210 | 210 | |||
Seller 1 | 180 | 200 | |||
Transporter 2 | Seller 2 | 200 | 1200 | 210 | 1000 |
Seller 3 | 210 | 220 |
Transporter 1 | Transporter 2 | ||||
Mode of generalized transport | unit cost($/t) | maximum amount(t) | unit cost($/t) | maximum amount(t) | |
Transporter 1 | Supplier 1 | 180 | 2000 | 190 | 2500 |
Transporter 2 | Supplier 2 | 210 | 2200 | 220 | 2000 |
Transporter 1 | Transporter 2 | ||||
Mode of generalized transport | unit cost($/t) | maximum amount(t) | unit cost($/t) | maximum amount(t) | |
Transporter 1 | Supplier 1 | 180 | 2000 | 190 | 2500 |
Transporter 2 | Supplier 2 | 210 | 2200 | 220 | 2000 |
Seller 1 | Seller 2 | Seller 3 | |
Unit product sales cost ($/t) | 80 | 90 | 85 |
Product demand (t) | 1000 | 1200 | 800 |
Seller 1 | Seller 2 | Seller 3 | |
Unit product sales cost ($/t) | 80 | 90 | 85 |
Product demand (t) | 1000 | 1200 | 800 |
(0 0 800 0 1000 200 0 0 0 0 1000 0) | (0.556 0 0.444 0) | 1171.8 |
(0 0 800 0 1000 200 0 0 0 0 1000 0) | (0.556 0 0.444 0) | 1171.8 |
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