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A class of polynomially solvable 0-1 programming problems and applications

This paper explores the concept of Early Order Commitment (EOC) in supply chains, analyzing its impact on retailers and manufacturers through an analytical model and simulation studies. It delves into cost savings, supply chain optimization, uncertainty management, and decision-making processes under varying conditions. The study provides insights into the benefits and challenges of implementing EOC strategies in supply chain management.

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A class of polynomially solvable 0-1 programming problems and applications

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  1. A class of polynomially solvable 0-1 programming problems and applications Jinxing Xie (谢金星) Department of Mathematical Sciences Tsinghua University, Beijing 100084, China E-mail:jxie@math.tsinghua.edu.cn http://faculty.math.tsinghua.edu.cn/~jxie 合作者:赵先德,魏哲咏,周德明 王 淼,熊华春,邓晓雪

  2. Outline • Background: Early Order Commitment • An Analytical Model: 0-1 Programming • A Polynomial Algorithm • Other Applications

  3. Connect Supply With Demand: The most important issue in supply chain management (SCM) Information SUPPLY DEMAND Product Cash Supply chain optimization & coordination (SCO & SCC): The members in a supply chain cooperate with each other to reach the best performance of the entire chain

  4. Supply Chain Coordination:Dealing with Uncertainty • Uncertainty indemand and leadtime (提前期) • Leadtime reduction: time-based competition SUPPLY DEMAND • Make to stock • Make to order

  5. Supply Chain Coordination:Dealing with Uncertainty • Information sharing– sharing real-time demand data collected at the point-of-sales with upstream suppliers (e.g., Lee, So and Tang (LST,2000); Cachon and Fisher 2000; Raghunathan 2001; etc.) • Centralized forecasting mechanism – CPFR • Contract design – coordinate the chain • ……

  6. Early Order Commitment (EOC) • means that a retailer commits to purchase a fixed-order quantity and delivery time from a manufacturer before the real need takes place and in advance of the leadtime. (advance ordering/booking commitment) • is used in practice for a long time, e.g. by Walmart • is an alternative form of supply chain coordination (SCC)

  7. EOC: Questions • Why should a retailer make commitment with penalty charge? • Intuition: EOC increases a retailer’s risks of demand uncertainty, but helps the manufacturer reduce planning uncertainty • Our work • Simulation studies • Analytical model for a supply chain with infinite time horizon

  8. EOC: Simulation Studies • Zhao, Xie and Lau (IJPR2001), Zhao, Xie and Wei (DS2002), Zhao, Xie and Zhang (SCM2002), etc. conducted extensive simulation studies under various operational conditions. • Findings • EOC can generate significant cost savings in some cases • Can we have an analytical model? (Zhao, Xie and Wei (EJOR2007), Xiong, Xie and Wang (EJOR2010), etc.)

  9. Supplier (Manufacturer) Demand Retailer Basic Assumptions: Same asLST(MS, 2000) • The demand is assumed to be a simple autocorrelated AR(1) process • d > 0, -1<<1, and  is i.i.d. normally distributed with mean zero and variance 2. • << d  negative demand is negligible

  10. Notation • L - manufacturing (supplier) leadtime • l - delivery leadtime • A - EOC period 0 <= A <= L+1 • Further (techinical) assumptions: • An “alternative” source exists for the manufacturer • Backorder for the retailer • No fixed ordering cost • Information sharing between the two partners A l Order Delivery leadtime

  11. An order and delivery flow • PT = L, DT = l, EOCT = A (decision)

  12. Time Label t-A t-A+1 t t+1 t+A t+l+A+1 Retailer’s Demand Dt Dt+1Dt+l+A+1 Retailer’s Order Ot-AOt-A+1 Ot Ot+L-A+1 Manufacturer’s Demand D’tD’t+1D’t+A D’t+L+1 Manufacturer’s Order Qt Time Label t t+1 t+A t+L-A+1 t+L+1 Framework of Decision Making : Periodic-review (at end of each period)

  13. Retailer’s Ordering Decision (1) • the total demand during periods [t+1, t+l+A+1]

  14. Retailer’s Ordering Decision (2) • the order-up-to level (optimal) • retailer’s order quantity at period t

  15. Manufacturer’s Ordering Decision (1) • Manufacturer’s demand for [t+1, t+L+1] is

  16. Manufacturer’s Ordering Decision (2)

  17. Manufacturer’s Ordering Decision (3) • The order-up-to level (optimal) • order quantity at period t is

  18. Cost Measures • Retailer’s average cost per period • Manufacturer’s average cost per period • total cost of the supply chain Normal Loss Function

  19. Supply Chain’s Relative Cost Saving “Cost Ratio” • Critical condition when EOC is beneficial

  20. How ∆SC changes with A? • Theorem.∆SC decreases at first and then increases as A increases from 0 to L+1. • Corollary. The optimal A* = 0 or L+1. • Managerial implications -- Either do not use EOC policy (make to stock) or use the largest possible EOC periods (make to order)

  21. Performance of EOC: Example(τ=1.0, l=6, L=12, =0.5)

  22. Note on τ: usually, τ 1 Observation.(H+P)η(x)+Hxis convex in xand its minimum is achieved at K Usually: h  H, p  P(h+p)η(k)+hk  (H+P)η(K)+HK  under most situations in practice, cost ratio τ 1

  23. How τ, l, Linfluence the performance of EOC? • Proposition 1. When τ1, EOC is always beneficial. • Proposition 2. When τ>1, as r increases, the critical condition is getting difficult to hold. • Proposition 3. When τ>1, as L increases, the critical condition is getting difficult to hold. • Proposition 4. When τ>1 and , as l increases, (LHS – RHS) of the critical condition inequality increases at first and then decreases.

  24. EOC: Multiple retailers • i=1, 2, …, n:

  25. EOC: 0-1 programming • i=1, 2, …, n: xi=0,or xi=L+1 • Similar to previous analysis:

  26. EOC: 0-1 programming • Theorem

  27. EOC: Algorithm • 算法:

  28. EOC: generations • From 2-stage to more stages

  29. i=1 m m-1 j ...... 1 ...... Cmi Cm-1,i Cji C1i Stage Component Base-assembly End Product Other applications • Single period problem: commonality decision in a multi-product multi-stage assembly line • For each stage j: commonality Cjc with ...... i=n

  30. Commonality decision • Assumptions: salvage=0; stockout not permitted • Turn to spot market: the purchasing cost of the component Cji is eji (i=1,2,…,n,c ; j= m,m-1,…,1) • assume ejc ≥ eji > cji (i=1,2,…,n; j= m,m-1,…,1) • Decisions: • Whether dedicated component Cji should be replaced by the common components Cjc • Inventory levels for all components Cji (i=1,2,…,n,c ; j= m,m-1,…,1)

  31. Commonality decision • Objective function (expected profit)

  32. Commonality decision • Denote • Proposition. Suppose that the component commonality decision is given, then

  33. Two different cases • Case (a) (Component commonality): • The component commonality decisions in a stage are independent of those in other stages. • Case (b) (Differentiation postponement): • The dedicated component Cji can be replaced by the common component Cjc only if the dedicated components Cj+1,i , Cj+2,i,…,Cmi are replaced by Cj+1,c , Cj+2,c ,…,Cmc (i.e., , for any and i=1,2,…,n).

  34. Case (a) • 0-1 Programming which can be decoupled into m sub-problems (for j ) • In an optimal solution:

  35. Case (a) • rji be the ranking position of bji among {bj1, bj2, … , bjn} O(mn2)

  36. Case (b) • 0-1 programming • Enumeration method: • An algorithm with complexity

  37. Other applications? • Basic patterns: square-root function + linear function • Risk management?

  38. Thanks for your attendance!

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