Leadtime-inventory trade-offs in assemble-to-order systemsObjective of the paperMethodology and ResultsOverviewOverviewSimplest Model : intuitive result (1/2)Simplest Model : intuitive result (2/2)OverviewGeneral Model : Objective and notationsGeneral Model – AssumptionsOverviewGeneral single item ModelFIRST THEOREM :FIRST THEOREM : interpretationSingle product (multi item) ModelSingle product (multi item) ModelSingle product (multi item) ModelSECOND THEOREM :General Model with Poisson ordersSolutionTHIRD THEOREMOverviewSingle-Item Systems approximationSingle-Item Systems approximationMultiple-Item Systems approximationMultiple-Item Systems approximationOverviewConclusionLeadtime-inventory trade-offs in assemble-to-order systemsbyPaul Glasserman and Yashan Wang15.764 The Theory of Operations ManagementMarch 02, 2004Presentation by Nicolas MiegevilleThis is a summary presentation based on: Wang, Y., and P. Glasserman. "Lead-time Inventory Trade-offs in Assemble-to-order Systems." Operations Research 43, no. 6 (1998).Objective of the paper7/6/2004 15.764 The Theory of Operations Management 2 Usual qualitative statement : inventory is the currency of service.  Operations Management books Management reviews Research papers May we find a quantitative measure of the marginal cost of a service improvement in units of inventory ?Methodology and Results7/6/2004 15.764 The Theory of Operations Management 3 We focus on a particular class of models : assemble-to-order models with stochastic demands and production intervals items are made to stock to supply variable demands for finished products Multiple FP are ATO from the items (one product may contend several times the same item) For each item : continuous-review base-stock policy : one demand for a unit triggers a replenishment order Items are produced one at time on dedicated facilities We measure the service by the fill rate : proportion of orders filled before a target (delivery leadtime). We prove that there is a LINEAR trade-off between service and inventory, at high levels of service.Overview7/6/2004 15.764 The Theory of Operations Management 4 Intuitive simplest model General Model Three theorems for three models Single item model Single product multi item model Multi product multi item model Checking the Approximations given by the theorems ConclusionOverview7/6/2004 15.764 The Theory of Operations Management 5 Intuitive simplest model General Model Three theorems for three models Single item model Single product multi item model Multi product multi item model Checking the Approximations given by the theorems ConclusionSimplest Model : intuitive result (1/2)7/6/2004 15.764 The Theory of Operations Management 6 Single one item-product model  Orders arrive in a Poisson stream (rate ) Time to produce one unit is exponentially distributed (mean ) s denote the base-stock level x denote the delivery time R denote the steady-state (we assume ) response time of an order Queuing Theory : M/M/1 results At a fixed fill rate :λµ<1µ()()1()1sxPR x PR x eµλλµ−−⎛⎞≤=− >=−⎜⎟⎝⎠λ[](1 ) 0,1δ−∈ln( )ln( ) ln( )sxδµλλµµλ⎛⎞⎛⎞⎜⎟⎜⎟−⎜⎟=−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠Simplest Model : intuitive result (2/2)7/6/2004 15.764 The Theory of Operations Management 712310δδδ;; ;;3δx∆ln( )ln( ) ln( )sxδµλλµµλ⎛⎞⎛⎞⎜⎟⎜⎟−⎜⎟=−⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠xs2δ1δLevel curves of constant service are stright linesof constant slopeService increasess∆3δFor a fixed rate, increase of base-stockdecreases the delivery leadtimeOverview7/6/2004 15.764 The Theory of Operations Management 8 Intuitive simplest model General Model Three theorems for three models Single item model Single product multi item model Multi product multi item model Checking the Approximations given by the theorems ConclusionGeneral Model : Objective and notations7/6/2004 15.764 The Theory of Operations Management 9 Multiple items are produced on dedicated facilities and kept in inventories A product is a collection of a possible RANDOM number of items of each type (~components) The assembly operation is uncapacitated Notations : A is the order interarrival time (products). B is the unit production interval (items) D is the batch order size (items per product)  R is the response time  s is the base-stock level (items) x is the delivery timeGeneral Model – Assumptions7/6/2004 15.764 The Theory of Operations Management 10i=1…d itemsj=1…m products(See Figure 1, page 859 of the Glassermanand Wang paper.)We assume the production intervals (B), the interarrival times (A) and the batch size vectors (D) are ALL INDEPENDENTS of each other.Overview7/6/2004 15.764 The Theory of Operations Management 11 Intuitive simplest model General Model Three theorems for three models Single item model Single product multi item model Multi product multi item model Checking the Approximations given by the theorems ConclusionGeneral single item Model7/6/2004 15.764 The Theory of Operations Management 12 For any r.v. Y, we note the cumulant generating function : Let’s introduce the r.v. : We have :() ln([ ])YYEeθψθ=1DjjXBA==−∑[] [].[] []EXEBEDEA=−2[ ] [ ]. [ ] [ ].( [ ]) [ ]Var X E D Var B Var D E B Var A=+ +() ( ()) ( )XDB Aψθψψθ ψ θ=+−We require E[X]<0 s.t. the steady-stateresponse time exists(1)(0) [ ]YEYψ=(2)(0) [ ]YVar Yψ=7/6/2004 15.764 The Theory of Operations Management 13If it exists, it is unique (f Convex and f(0)=0)FIRST THEOREM : If there is a at which , then with For a level curve of constant service, we have approximately : Proof of theorem uses the concept of associated queue to the response time (Lemma 1), in which we can show (if the system is stable) by using the Theorem of Gut (1988) a convergence in distribution of the waiting time. Then an exponential twisting gives the result (Wald’slikelihood ratio identity). With an assembly time Un (random delay iid and bounded), the result is still true (() )xssxlim e P R s x Cγβ++→∞>=() 0Xψγ=0γ>()Bβψγ=with C constant >01ln( )Csxγββδ=− +FIRST THEOREM : interpretation7/6/2004 15.764 The Theory of Operations Management 14 For a level curve of constant service, one unit increase