# Order Full Rate, Leadtime Variability, & Advance Demand Information In An Assemble To Order System - Ping Xu

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Published on February 4, 2009

Author: siddharth4mba

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Order Full Rate, Leadtime Variability, & Advance Demand Information In An Assemble To Order System - Ping Xu

Order Full Rate, Leadtime Variability, and Advance Demand Information in an Assemble- To-Order System by Lu, Song, and Yao (2002) Presented by Ping Xu This summary presentation is based on: Lu, Yingdong, and Jing-Sheng Song. quot;Order-Based Cost Optimization in Assemble-to-Order Systems.quot; To appear in Operations Research, 2003. 7/6/2004 Ping Xu 1

Preview • Assembly-to-order system – Each product is assembled from a set of components, – Demand for products following batch Poisson processes, – Inventory of each component follows a base-stock policy – Replenishment leadtime i.i.d. random variables for each component. Model as a M / G / ∞ X • queue, driven by a common multiclass batch Poisson input stream – Derive the joint queue-length distribution, – Order fulfillment performance measure. 7/6/2004 Ping Xu 2

Model • M different components, and F = {1,2,…,m} are the component indices. Customer orders arrive as a stationary Poisson process, {A(t), t>=0}, with rate λ . • • Order type K: it contains positive units of component in K and 0 units in FK. An order is of type K with probability qk, ∑ K q = 1 K – Type K order stream forms a compound Poisson process with rate λ = q λ K K – A type-K order has Q j units for each component j, QK = (Q j , j ∈ K ) has a known K K – discrete distribution. • For each component i, the demand process forms a compound Poisson process. 7/6/2004 Ping Xu 3

Model • Demand are filled on a FCFS basis. • Demand are backlogged (if one or more components are missing), and are filled on a FCFS basis. • Inventory of each component is controlled by an independent base-stock policy, – Where si is the base-stock level for component i – For each component i, replenishment leadtimes, Li , are i.i.d., with a cdf of Gi Net inventory at time t, I i ( t ) = s i − X i ( t ), i = 1, … m – , where Xi(t) is the number of outstanding orders of component i at time t. • Immediate availability of all components needed for an arriving demand as the “off-the-shelf” fill rate. – Off-the-shelf fill rate of component i, f i = P[ X i + Qi ≤ si ] – Off-the-shelf fill rate of demand type K, f = P[ X i + Qi ≤ si , ∀i ∈ K ] K K – Average (over all demand types) off-the-shelf fill rate, f = ∑ q K f K K 7/6/2004 Ping Xu 4

Performance Analysis X (t ) = ( X 1 (t ),… , X m (t )) • Derive the joint distribution and steady state limit of vector (See “Suppliers/Arrivals Replenishment Orders” diagram in Lu, Song, and Yao paper) – Each component i, the number of outstanding orders is exactly the number of jobs in service in an M iQi / Gi / ∞ queue with Poisson arrival λi and batch size Qi – The m queues are not independent. Given the number of demand arrivals up to t, the X i (t ) s are independent of one another. – 7/6/2004 Ping Xu 5

Performance Analysis Proposition 1: X (t ) = ( X 1 (t ),… , X m (t )) has a limiting distribution. Derive the generating • function of X. In the special case of unit arrival, Qi ≡ 1 , the generating function of X corresponds to a • multivariate Poisson distribution. For each i, X i is a Poisson variable with parameter λi i = (Σ K∈ℜi λ K ) i • The correlation of the queue is solely induced by the common arrivals. If the proportion of the demand types that require both i and j are very small, the correlation between Xi and Xj is negligible. • Level of correlation is independent of the demand rate. • Reducing the variability of leadtime or batch sizes will result in a higher correlation among the queue lengths of outstanding jobs. 7/6/2004 Ping Xu 6

Response-time-based order fill rate f K ( w) is the probability of having all the components ready within w units of time. 1) Di (t , t + u ] := Di (t + u ) − Di (t ) 2) Total number of departures from queue i in (τ ,τ + w) = X i (τ ) + Di (τ ,τ + w] − X i (τ + w) 3) I i (τ ) + { X i (τ ) + Di (τ ,τ + w] − X i (τ + w)} ≥ 0 4) X i (τ + w) − Di (τ ,τ + w] ≤ si , i ∈ K 5) QiK X i (τ + w) = X (τ ) + ∑1{Ln > w} + X i (τ ,τ + w] w i i 6) n =1 can be supplied by τ + w iff 7) Demand at QiK X iw (τ ) + X i (τ ,τ + w] − Di (τ ,τ + w] ≤ si − ∑1{Ln > w}, i ∈ K i n =1 Yi := X iw − Yi w ⎡ ⎤ K Qi Order fill rate of type-K demand within time window w, f ( w) = P ⎢Yi + ∑ 1{Li > w} ≤ si , ∀i ∈ K ⎥ Kn 8) ⎢ ⎥ ⎣ ⎦ n =1 ⎛ ⎞ 9) Mean: E[Yi ] = ⎜ ∑ λ E (Qi ) ⎟ ( i − w) ℑ ℑ ⎜ ⎟ ⎝ ℑ∈ℜi ⎠ 7/6/2004 Ping Xu 7

Connection to advance demand information • Suppose each order arrival epoch is known w time units in advance, where w>0 is a deterministic constant. • Suppose a type-K order arrives at , and this information is known at , we can fill this order upon its arrival with probability, • Advance demand information improves the off-the-shelf fill rate: ˆ Compare f AK (0) with that of the modified system, f K (0) , where leadtime is reduced from • Li to Li = [ Li − w]+ ˆ ⎧ ⎫ K ˆ K (0) = P ⎪ X w + 1[ Ln > 0] ≤ s , ∀i ∈ K ⎪ ≤ f K ( w) Qi ⎨i ∑ i ˆ ⎬ f i ⎪ ⎪ ⎩ ⎭ n =0 • Knowing demand in advance (by w time units) is more effective, in terms of order fill rate, than reducing the supply leadtime of components. 7/6/2004 Ping Xu 8

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