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A class of polynomially solvable 0-1 programming problems and applications Jinxing Xie ( ) Department of Mathematical Sciences Tsinghua University, Beijing 100084, China E-mail:jxie@math.tsinghua.edu.cn http://faculty.math.tsinghua.edu.cn/~jxie Slide 2 Outline Background: Early Order Commitment An Analytical Model: 0-1 Programming A Polynomial Algorithm Other Applications Slide 3 Connect Supply With Demand: The most important issue in supply chain management (SCM) SUPPLYDEMAND Information 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 Slide 4 Supply Chain Coordination: Dealing with Uncertainty Uncertainty in demand and leadtime ( ) Leadtime reduction: time-based competition DEMANDDEMAND SUPPLYSUPPLY Make to stock Make to order Slide 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 Slide 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) Slide 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 Slide 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.) Slide 9 Basic Assumptions: Same as LST(MS, 2000) The demand is assumed to be a simple autocorrelated AR(1) process d > 0, -1< How , l, L influence the performance of EOC? Proposition 1. When 1, EOC is always beneficial. Proposition 2. When >1, as r increases, the critical condition is getting difficult to hold. Proposition 3. When >1, as L increases, the critical condition is getting difficult to hold. Proposition 4. When >1 and, as l increases, (LHS RHS) of the critical condition inequality increases at first and then decreases. Slide 24 EOC: Multiple retailers i=1, 2, , n: Slide 25 EOC: 0-1 programming i=1, 2, , n: x i =0 or x i =L+1 Similar to previous analysis: Slide 26 EOC: 0-1 programming Theorem Slide 27 EOC: Algorithm : Slide 28 EOC: generations From 2-stage to more stages Slide 29 Other applications Single period problem: commonality decision in a multi-product multi-stage assembly line For each stage j: commonality C jc with C1iC1i C ji C m-1,i C mi m-1m...... j 1 i=1 StageComponent Base-assembly End Product i=n...... Slide 30 Commonality decision Assumptions: salvage=0; stockout not permitted Turn to spot market: the purchasing cost of the component C ji is e ji (i=1,2,,n,c ; j= m,m-1,,1) assume e jc e ji > c ji (i=1,2,,n; j= m,m-1,,1) Decisions: Whether dedicated component C ji should be replaced by the common components C jc Inventory levels for all components C ji (i=1,2,,n,c ; j= m,m-1,,1) Slide 31 Commonality decision Objective function (expected profit) Slide 32 Commonality decision Denote Proposition. Suppose that the component commonality decision is given, then Slide 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 C ji can be replaced by the common component C jc only if the dedicated components C j+1,i, C j+2,i,,C mi are replaced by C j+1,c, C j+2,c,,C mc (i.e.,, for any and i=1,2,,n). Slide 34 Case (a) 0-1 Programming which can be decoupled into m sub-problems (for j ) In an optimal solution: Slide 35 Case (a) r ji be the ranking position of b ji among {b j1, b j2, , b jn } O(mn 2 ) Slide 36 Case (b) 0-1 programming Enumeration method: An algorithm with complexity Slide 37 Other applications? Basic patterns: square-root function + linear function Risk management? Slide 38 Thanks for your attendance!