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# Filled function method to optimize supply chain transportation costs

• * Corresponding author: Youlin Shang

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.

Mathematics Subject Classification: 90C30.

 Citation: • • Table 1.

 The proposed Algorithm Algorithm in  $f({x^*})$ $f({x^*})$ 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  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

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

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

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
• Tables(6)

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