Optimal Inventory Policies for Assembly Systems under Random DemandsMain ResultMain ResultRelevant LiteratureModelModelNotationNotation: contd..Model: Cost parametersModel: CostLong-Run Inventory PositionLong-Run BalanceAssumptions on Cost parametersLong-run Inventory positionLong-run Inventory positionEquivalent Series SystemEquivalent Series SystemNew Formulation for PEquivalent Series SystemEquivalent Series SystemGeneral Assumption on CostsGeneralized AssumptionPractical Necessity of Generalized AssumptionSummaryCommentsOptimal Inventory Policies for Assembly Systems under Random DemandsKaj RoslingPresented by:Shobhit Gupta, Operations Research CenterThis summary presentation is based on: Rosling, K. “Optimal Inventory Policies for Assembly Systems Under Random Demand.” Operations Research 43 (6), 1989.Main Result• Remodel an assembly system as a series system• Simple re-order policies are optimal (See Figure 1 on page 566 of the Rosling paper)(See Figure 2 on page 571 of the Rosling paper)Main Result• Assembly system:- Ordered amounts available after a fixed lead time- Random customer demands only for the end product- Assumptions on cost parameters• Under assumptions and restriction on initial stock levels, assembly system can be treated as a series system• Optimal inventory policy – can be calculated by approach in Clark and Scarf’s paper (1960)**Clark, A.J. and Herbert Scarf. “Optimal Policies for a Multi-echelon InventoryProblem.” Management Science 6 (1960): 475-90.Relevant Literature• Clark and Scarf (1960) – derive optimal ordering policy for pure series system• Fukuda (1961) – include disposal of items in stock• Federgruen and Zipkin (1984) – generalize Clark and Scarf approach to stationary infinite horizon caseModel• N items (components, subassemblies, the end item)• Each non-end item has exactly one successor- product networks forms a tree rooted in the end item• Exactly one unit of each item required for the end item• Notation:= unique immediate successor of item i=1…N;= the set of all successors of item i= the set of immediate predecessors of item i= the set of all predecessors of item I= number of periods (lead-time) for assembly (or delivery) of item i)(is0)1(=s)(iA)(iP)(iBilModel• At the beginning of a time period:1. Outstanding orders arrive and new ordering decisions made2. Old backlogs fulfilled and customer demands occur (for the end period)3. Backlog and inventory holding costs incurredNotation= iid demand in period t for the end item with density and distribution = , expected value of = echelon inventory position of item i in period t before ordering decision are made ( = inventory on hand + units in assembly/order - units backlogged)= echelon inventory position of item i in period t after ordering decisions are made;= amount ordered for item i in period t;arrives after periodsitX][tEξtξitYtξλititXY ≥ititXY −il)(⋅φ)(⋅ΦNotation: contd..• = echelon inventory on hand of item i in period t before ordering decisions are made but after assembles arrive= • cannot order more than at hand(no intermediate shortage)litX∑−−=−−1,tltkkltiiiYξ)( if kPiXYlitkt∈≤Model: Cost parameters= unit installation holding cost per period of item i = unit echelon holding cost per period of item i= unit backlogging cost per period of the end item= period discount factor • Cost in period t cost gBacklogginitem/ endfor cost Holding iHih∑∈−=)(iPkkiiHHhpα10 ≤<α),0(Max),0(Max)(111)(2ltttltltislitNiiXpXHXXH −⋅+−⋅+−∑=ξξCost Holdingon InstallatiModel: Cost• Alternate Formulation:• Using • Total Expected Cost over an infinite horizon:(see equation 1 on page 567): convolution of over periods)(11⋅Φ+l)(⋅Φ)1(1+l∑+=++−=−iiiltssitltlltiYX1,ξξ(see page 567 of Rosling paper, left hand column)Long-Run Inventory Position• : total lead-time for item i and all its successors • (See page 567, part 2 “Long-Run Inventory Position)∑∈+=)(iAkkiillMiMLong-Run Balance• Assembly system is in long-run balance in period t iff for i=1,..,N-1• Inventory positions equally close to the end item increase with total lead-time• Satisfied trivially if 1,...,1for ,1−=≤−+−iMtiMitMXXµµµ)()1( iPi∈+Assumptions on Cost parameters•– All echelon holding costs positive •– Better to hold inventory than incur a backlog allfor 0 ihi>11)(HphisMNii+<⋅−=∑αLong-run Inventory positionLemma 1: (See page 568 of Rosling paper)Lemma 2: (See page 568 of Rosling paper)Long-run Inventory positionTheorem 1: “Any policy satisfying Lemmas 1 and 2 leads the system into long-run balance and keeps it there. This will take not more than MN+1 periods after accumulated demand exceeds Maxi Xi1.”Proof: Outline––– long-run balance for i for – Upper bound q(i)1 Lemmaby )( allfor ,1iqtXXLtiit≥≤+µξξµµ−+==−≤−=−+−=−=+−∑∑iMtitqrrtqrLqiriqMitMiqtXXYX)(for ,111,1iMiqt+≥ )(Equivalent Series SystemTheorem 2: If the Assumptions hold and system is initially in long-run balance, then optimal policies of the assembly system are equivalent to those of a pure series system where:– i succeeds item i+1– lead-time of item i is Li– holding costProof: Cost functioniiLliihh−⋅←αConstant )()()()(E Min11111i1Y1+⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−+⋅+⋅⋅∫∑∑∞+=−∞=−ξξφξααααdYHpYhitiiiYLitLNiitLliLtEquivalent Series SystemEasy to show, using Theorem 1,:Hence, using Lemma 1,Use this constraint in Problem P.)( and , allfor iPktiXXiMMktit∈≤−LtiititXYX,1*+≤≤New Formulation for PConstant )()()()(E Min11111i1Y1+⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛−+⋅+⋅⋅∫∑∑∞+=−∞=−ξξφξααααdYHpYhitiiiYLitLNiitLliLttiXYXLtiitit, allfor such that,1+≤≤11,1,11andwhere−−−−=−++−=−=∑ttiittLtssLtiL,tiYXYXξξEquivalent Series SystemCorollary 2: There exist Si’s such that the following policy is optimal for all i and tSi– obtained from Clark and Scarf’s (1960) procedure • Critically dependent on initial inventory level assumption (long-run balance initial inventory levels)• Generally optimal policy by Schmidt and Nahmias (1985)iitititiitLtiiitSXXYSXXSY≥=≤=+ if if