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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

* Corresponding author: Youlin Shang

Received  October 2020 Revised  February 2021 Early access July 2021

Fund Project: The first author is supported by NSF grant of China (Nos.12071112, 11701150, 11471102); Basic research projects for key scientific research projects in Henan Province of China (No.20ZX001)

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.

Citation: Deqiang Qu, Youlin Shang, Dan Wu, Guanglei Sun. Filled function method to optimize supply chain transportation costs. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021115
##### 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.

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##### 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] $f({x^*})$ $f({x^*})$ 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] $f({x^*})$ $f({x^*})$ 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 to seller unit cost and maximum transport volume
 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 to supplier unit cost and maximum transport volume
 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's unit product cost and demand
 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
 ${x^*}$ ${\beta ^*}$ $f({x^*})$ (0 0 800 0 1000 200 0 0 0 0 1000 0) (0.556 0 0.444 0) 1171.8
 ${x^*}$ ${\beta ^*}$ $f({x^*})$ (0 0 800 0 1000 200 0 0 0 0 1000 0) (0.556 0 0.444 0) 1171.8
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