• Previous Article
    First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits
  • JIMO Home
  • This Issue
  • Next Article
    Risk minimization inventory model with a profit target and option contracts under spot price uncertainty
doi: 10.3934/jimo.2021115
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Y. LiB. S. Han and Y. J. Yang, A novel one parameter filled function, Comm. on Appl. Math. and Comput, 24 (2010), 17-24.   Google Scholar

[5]

J. R. LiY. 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.  Google Scholar

[6]

H. W. LinY. L. GaoX. 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.  Google Scholar

[7]

Y. L. Shang, Research on Filled Function Method in Nonlinear Global Optimization, Ph.D thesis, Shanghai University in Shanghai of China, 2005. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

W. X. WangY. L. Shang and D. Wang, Filled function method for solving non-smooth box constrained global optimization problems, Operational Research Transactions, 23 (2019), 28-34.   Google Scholar

[11]

W. X. WangY. 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.   Google Scholar

[12]

Y. WangW. 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.  Google Scholar

[13]

Y. J. YangM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. J. YangZ. 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.  Google Scholar

[17]

L. YuanZ. 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.  Google Scholar

[18]

L. YuanZ. WanJ. 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.  Google Scholar

[19]

Y. ZhangL. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Y. LiB. S. Han and Y. J. Yang, A novel one parameter filled function, Comm. on Appl. Math. and Comput, 24 (2010), 17-24.   Google Scholar

[5]

J. R. LiY. 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.  Google Scholar

[6]

H. W. LinY. L. GaoX. 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.  Google Scholar

[7]

Y. L. Shang, Research on Filled Function Method in Nonlinear Global Optimization, Ph.D thesis, Shanghai University in Shanghai of China, 2005. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

W. X. WangY. L. Shang and D. Wang, Filled function method for solving non-smooth box constrained global optimization problems, Operational Research Transactions, 23 (2019), 28-34.   Google Scholar

[11]

W. X. WangY. 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.   Google Scholar

[12]

Y. WangW. 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.  Google Scholar

[13]

Y. J. YangM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. J. YangZ. 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.  Google Scholar

[17]

L. YuanZ. 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.  Google Scholar

[18]

L. YuanZ. WanJ. 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.  Google Scholar

[19]

Y. ZhangL. 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.  Google Scholar

Table 1.   
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
Table 2.   
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
Table 3.  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
Table 4.  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
Table 5.  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
Table 6.   
$ {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
[1]

Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911

[2]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353

[3]

Joseph Geunes, Panos M. Pardalos. Introduction to the Special Issue on Supply Chain Optimization. Journal of Industrial & Management Optimization, 2007, 3 (1) : i-ii. doi: 10.3934/jimo.2007.3.1i

[4]

Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial & Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727

[5]

Xiaohui Ren, Daofang Chang, Jin Shen. Optimization of the product service supply chain under the influence of presale services. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021130

[6]

Nina Yan, Tingting Tong, Hongyan Dai. Capital-constrained supply chain with multiple decision attributes: Decision optimization and coordination analysis. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1831-1856. doi: 10.3934/jimo.2018125

[7]

Qiong Liu, Ahmad Reza Rezaei, Kuan Yew Wong, Mohammad Mahdi Azami. Integrated modeling and optimization of material flow and financial flow of supply chain network considering financial ratios. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 113-132. doi: 10.3934/naco.2019009

[8]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

[9]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[10]

Abdolhossein Sadrnia, Amirreza Payandeh Sani, Najme Roghani Langarudi. Sustainable closed-loop supply chain network optimization for construction machinery recovering. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2389-2414. doi: 10.3934/jimo.2020074

[11]

Ziyuan Zhang, Liying Yu. Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021155

[12]

Jianbin Li, Niu Yu, Zhixue Liu, Lianjie Shu. Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1587-1611. doi: 10.3934/jimo.2016.12.1587

[13]

Amin Aalaei, Hamid Davoudpour. Two bounds for integrating the virtual dynamic cellular manufacturing problem into supply chain management. Journal of Industrial & Management Optimization, 2016, 12 (3) : 907-930. doi: 10.3934/jimo.2016.12.907

[14]

Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066

[15]

Lianxia Zhao, Jianxin You, Shu-Cherng Fang. A dual-channel supply chain problem with resource-utilization penalty: Who can benefit from sales effort?. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2837-2853. doi: 10.3934/jimo.2020097

[16]

Fatemeh Kangi, Seyed Hamid Reza Pasandideh, Esmaeil Mehdizadeh, Hamed Soleimani. The optimization of a multi-period multi-product closed-loop supply chain network with cross-docking delivery strategy. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021118

[17]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048

[18]

Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485

[19]

Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169

[20]

Honglin Yang, Jiawu Peng. Coordinating a supply chain with demand information updating. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020181

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (40)
  • HTML views (102)
  • Cited by (0)

Other articles
by authors

[Back to Top]