A class of polynomially solvable 0-1 programming problems and applications Jinxing Xie ( 谢金星 ) Department of Mathematical Sciences Tsinghua University, Beijing , China 合作者：赵先德，魏哲咏，周德明 王 淼，熊华春，邓晓雪

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

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

Supply Chain Coordination: Dealing with Uncertainty Uncertainty in demand and leadtime ( 提前期 ) Leadtime reduction: time-based competition DEMANDDEMAND SUPPLYSUPPLY Make to stock Make to order

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 ……

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)

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

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.)

Basic Assumptions: Same as LST(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 Supplier (Manufacturer) Retailer Demand

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 Delivery leadtimeOrder A l

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

Manufacturer’s Order Q t Manufacturer’s Demand D’ t D’ t+1 D’ t+A D’ t+L+1 Retailer’s Demand D t D t+1 D t+l+A+1 Retailer’s Order O t-A O t-A+1 O t O t+L-A+1 Time Label t-A t-A+1 t t+1 t+A t+l+A+1 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)

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

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

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

Manufacturer ’ s Ordering Decision (2)

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

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

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

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)

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

Note on τ : usually, τ  1 Observation. (H+P)η(x)+Hx is convex in x and 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

How τ, l, L influence 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.

EOC: Multiple retailers i=1, 2, …, n:

EOC: 0-1 programming i=1, 2, …, n: x i =0 ， or x i =L+1 Similar to previous analysis:

EOC: 0-1 programming Theorem

EOC: Algorithm 算法 :

EOC: generations From 2-stage to more stages

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......

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)

Commonality decision Objective function (expected profit)

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

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).

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

Case (a) r ji be the ranking position of b ji among {b j1, b j2, …, b jn } O(mn 2 )

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

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

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