Information about Order Full Rate, Leadtime Variability, & Advance Demand Information In...

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

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

Canvas Prints at Affordable Prices make you smile.Visit http://www.shopcanvasprint...

30 Días en Bici en Gijón organiza un recorrido por los comercios históricos de la ...

Con el fin de conocer mejor el rol que juega internet en el proceso de compra en E...

With three established projects across the country and seven more in the pipeline,...

Retailing is not a rocket science, neither it's walk-in-the-park. In this presenta...

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

Read more

Order Full Rate, Leadtime Variability, ... Leadtime Variability, and Advance Demand Information in an ... Ping Xu 1 Preview? Assembly-to-order system – ...

Read more

"Lead-time Inventory Trade-offs in Assemble ... Full Rate, Leadtime Variability, and Advance Demand Information in an Assemble-To-Order System Author: Ping Xu

Read more

RESEARCH PAPER NO. 1811 A Separation Principle ... Order ﬁll rate, leadtime variability and advance demand information in an assemble-to-order system, ...

Read more

We consider an assemble-to-order (ATO) N-system, ... Order fill rate, lead-time variability and advance demand information in an Assemble-to-order system ...

Read more

Definition of lead time: Number of minutes, ... variable lead time order lead time effective lead time fixed lead time production lead time ...

Read more

Performance analysis and evaluation of ... lead-time variability and advance demand ... Dependence analysis of assemble-to-order systems - Xu ...

Read more

... multiple classes of demand, assemble-to-order system. ... fill rate, lead-time variability, and advance demand information in an assemble-to-order system.

Read more

This paper examines a two-tier assemble-to-order system. ... rate, leadtime variability and advance demand ... systems. Handbooks in Operations Research ...

Read more

... in continuous-review assemble-to-order (ATO) systems ... Yao D. D. Order fill rate, leadtime variability, and advance demand information in an assemble ...

Read more

## Add a comment