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Supply Chain Management Models with Quantity Discount Functions

Supply Chain Management Models with Quantity Discount Functions. Jung-Fa Tsai Department of Business Management, National Taipei University of Technology, Taiwan E-mail : jftsai@ntut.edu.tw. Outline. Introduction Problem formulation Linear strategies

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Supply Chain Management Models with Quantity Discount Functions

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  1. Supply Chain Management Models with Quantity Discount Functions Jung-Fa Tsai Department of Business Management, National Taipei University of Technology, Taiwan E-mail : jftsai@ntut.edu.tw

  2. Outline • Introduction • Problem formulation • Linear strategies • SCM models with various quantity discount functions • Conclusions

  3. Introduction • The average prices should not always be constant. To encourage buyers to order more, vendors usually offer quantity discounts. • This study solves a nonlinear supply chain management (SCM) model capable of treating various discount cost functions simultaneously, including linear, step, and multiple breakpoints cost functions. • A nonlinear model is converted into a linear model with linear strategies and the obtained solution is a global optimum.

  4. (Vendor) (Manufacturer) (Warehouse) (Distribution Center) (Customer) Vendor1 Customer1 Warehouse1 Manufacturer1 Distribution Center1 Vendor2 Customer2 Distribution Centern Manufacturern Vendorn Warehousen Customern Schema of an SCM model

  5. SCM model Minimize {product procurement costs + transportation costs + inventory costs} Subject to 1. Flow conservation. 2. Upper and lower bounds.

  6. SCM model Minimize s.t.

  7. Average price Average price P1 r1 P2 r2 P1 r rn-1 Pn-1 Pn Q Quantity Average price Qn-1 Qn Q1 Q2 Quantity Figure 1 A linear discount function Figure 3 A multiple breakpoint function P1 P2 P3 Q1 Q2 Q3 Q4 Quantity Figure 2 A step discount function Quantity discount functions

  8. SCM model with linear quantity discount function Min s.t. , other constraints. Since r is negative, the above program is a concave program solvable to obtain a global optimum with the piecewise linearization techniques.

  9. SCM model with step quantity discount function Min s.t. , other constraints. means cost level should be selected. The above program can be converted into a linear mixed 0-1 program as below.

  10. Transformed model Min s.t. other constraints.

  11. Multiple breakpoint quantity discount function Proposition 3 For a piecewise linear function depicted in Figure 3, can be expressed as: where

  12. SCM model with multiple breakpoint quantity discount function Min s.t. other constraints. The above program can be converted into a linear mixed 0-1 program as below.

  13. Transformed model Min s.t. other constraints.

  14. where is a linearization function of , and , are the break points of , ; and are the slopes of line segments between and , for j=1,2,…,m-1. Linear strategies (1) Proposition 1 A concave function can be piecewisely approximated as:

  15. , are the break points of … x Graphical Illustration

  16. Linear strategies (2) Proposition 2 A product term is equivalent to the following linear inequalities (i) ; (ii) . ,z is an unrestricted in sign variable, andis a large constant. Proof: If then , and if then .

  17. Manufacturer Warehouse Distribution Center A C E B D F Example The quantity discount functions are depicted with bold lines.

  18. Product cost (Pcost) and supplied limitation (Q1_up)

  19. Transportation cost from manufacturer to warehouse

  20. Inventory cost (Icost) and limitation (Q3_up)

  21. Transportation cost from warehouse to distribution center

  22. Distribution center demand (D)

  23. Manufacturer Warehouse Distribution Center 700/2 100/2 2000/42 1400 A C E 400/48 400/2 1300/1.7 1300/1.5 1000/2 1000 B D F “ / ” represents “quantity/unit cost” t=1 t=2 t=2 500/2 1800/42 500/2 1000 A C E 1000/1.5 1300/1.7 800 300/2 B D F t=2 t=3 t=3 Optimal results of the example

  24. Conclusions • Propose an SCM model capable of treating various discount cost functions such as linear, step, and multiple breakpoint functions; • The nonlinear SCM model can be converted into a linear model by linear strategies and then solved to obtain a global optimum, instead of obtaining a local optimum.

  25. Future research • Optimal Component Stocking Policy for Assemble-To-Order Systems • To determine how many and when to stock each of the components before the actual demand quantity is realized (at time zero) • Tradeoff between stocking too many components (cost) and too few (time) • Economic Order Quantity Model and Transportation Consideration (truck load)

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