Order Full Rate, Leadtime Variability, and Advance Demand Information in an Assemble- To-Order System

Transcription

1 Order Full Rate, Leadtme Varablty, and Advance Demand Informaton n an Assemble- To-Order System by Lu, Song, and Yao (2002) Presented by Png Xu Ths summary presentaton s based on: Lu, Yngdong, and Jng-Sheng Song. "Order-Based Cost Optmzaton n Assemble-to-Order Systems." To appear n Operatons Research, /6/2004 Png Xu

2 Preve Assembly-to-order system Each product s assembled from a set of components, Demand for products follong batch Posson processes, Inventory of each component follos a base-stock polcy Replenshment leadtme..d. random varables for each component. M X / G / Model as a queue, drven by a common multclass batch Posson nput stream Derve the jont queue-length dstrbuton, Order fulfllment performance measure. 7/6/2004 Png Xu 2

3 Model M dfferent components, and F = {,2,,m} are the component ndces. Customer orders arrve as a statonary Posson process, {A(t), t>=0}, th rate. Order type : t contans postve unts of component n and 0 unts n F\. An order s of type th probablty q k, q = Type order stream forms a compound Posson process th rate λ = q λ A type- order has Q j unts for each component j, Q = ( Qj, j ) has a knon dscrete dstrbuton. For each component, the demand process forms a compound Posson process. λ 7/6/2004 Png Xu 3

4 Model Demand are flled on a FCFS bass. Demand are backlogged (f one or more components are mssng), and are flled on a FCFS bass. Inventory of each component s controlled by an ndependent base-stock polcy, Where s s the base-stock level for component For each component, replenshment leadtmes, L, are..d., th a cdf of G Net nventory at tme t, I () t = s X (), t =, m, here X (t) s the number of outstandng orders of component at tme t. Immedate avalablty of all components needed for an arrvng demand as the off-the-shelf fll rate. f = P[ X + Q s ] Off-the-shelf fll rate of component, Off-the-shelf fll rate of demand type, f = P[ X + Q s, ] Average (over all demand types) off-the-shelf fll rate, f = q f 7/6/2004 Png Xu 4

5 Performance Analyss Derve the jont dstrbuton and steady state lmt of vector X ( t) = ( X ( t),, X ( t)) m (See Supplers/Arrvals Replenshment Orders dagram n Lu, Song, and Yao paper) Each component, the number of outstandng orders s exactly the number of jobs n servce n an Q M / G / queue th Posson arrval λ and batch sze Q The m queues are not ndependent. Gven the number of demand arrvals up to t, the are ndependent of one another. X () t s 7/6/2004 Png Xu 5

6 Performance Analyss X () t = ( X (), t, X ()) t Proposton : m has a lmtng dstrbuton. Derve the generatng functon of X. Q In the specal case of unt arrval,, the generatng functon of X corresponds to a multvarate Posson dstrbuton. For each, X s a Posson varable th parameter λ = ( Σ λ ) R The correlaton of the queue s solely nduced by the common arrvals. If the proporton of the demand types that requre both and j are very small, the correlaton beteen X and X j s neglgble. Level of correlaton s ndependent of the demand rate. Reducng the varablty of leadtme or batch szes ll result n a hgher correlaton among the queue lengths of outstandng jobs. 7/6/2004 Png Xu 6

7 Response-tme-based order fll rate ) f ( ) s the probablty of havng all the components ready thn unts of tme. 2) 3) Total number of departures from queue n 4) 5) 6) D (, tt+ u]: = D( t+ u) D() t I ( τ ) + { X ( τ) + D( τ, τ + ] X ( τ + )} 0 X ( τ + ) D( ττ, + ] s, Q X ( τ + ) = X ( τ) + { L > } + X ( τ, τ + ] n n= τ + (, τ τ + ) = X () τ + D(, τ τ + ] X ( τ + ) 7) Demand at can be suppled by ff Q n X ( τ) + X ( τ, τ + ] D( τ, τ + ] s { L > }, n= Y : = X Y 8) Order fll rate of type- demand thn tme ndo, 9) Mean: E[ Y] = λ I E( Q I ) ( ) I R Q n f ( ) = P Y + { L > } s, n= 7/6/2004 Png Xu 7

8 Connecton to advance demand nformaton Suppose each order arrval epoch s knon tme unts n advance, here >0 s a determnstc constant. Suppose a type- order arrves at, and ths nformaton s knon at, e can fll ths order upon ts arrval th probablty, Advance demand nformaton mproves the off-the-shelf fll rate: Compare f (0) th that of the modfed system, ˆ f (0), here leadtme s reduced from A Q ˆ (0) P [ ˆn f = X + L > 0] s, f ( ) n= 0 L to Lˆ = [ L ] + nong demand n advance (by tme unts) s more effectve, n terms of order fll rate, than reducng the supply leadtme of components. 7/6/2004 Png Xu 8