“Centralized Ordering Policies in a Multi-Warehouse System with Lead Times and Random Demand”System Description and AssumptionsWhy have a Depot? (with no inventory)Applicability of modelAllocation Assumption (for every period ordering, normal demand)The Allocation Assumption holds for high µ/σ and low NPolicy 1: Order Every Period (fixed ordering costs K = 0)Eppen and Schrage find an analytical expression for the inventory at each warehouseThe problem is now equivalent to the newsboy problem, and can be solved analyticallyPolicy 2: Order Up to level y every m periodsSeveral new assumptions lead to an analytical solution for the periodic ordering policy“Centralized Ordering Policies in a Multi-Warehouse System with Lead Times and Random Demand”A paper by Gary Eppen and Linus SchragePresentation by Tor SchoenmeyrThis is a summary presentation based on: Eppen, Gary, and Linus Schrage. “Centralized Ordering Policies in a Multi-Warehouse System with Lead times and Random Demand.” TIMS Studies in the Management Sciences, Vol. 16: Multi-Level Production/Inventory Control Systems, Theory and Practice. Edited by Leroy B. Schwarz. 1981.System and Problem Description The Allocation AssumptionPolicy 1: Order up to y every periodPolicy 2: Order up to y every m periodsSystem Description and AssumptionsDepot (no inventory)N warehouses(with inventories)SupplierCosts to be minimized:•Holding cost h per unit in inventory•Penalty cost p per unit of unmet demand (placed in backlog)•Fixed cost K for every order placedSupplier lead time LTransportationlead time lDemand(random)ZZZTotal inventory in system: y123111(,)eNµσ=222(,)eNµσ=333(,)eNµσ=Decisions to be made every period:•How much, if anything, should be ordered from the supplier•How should we distribute the incoming orders at the DepotWhy have a Depot? (with no inventory)Problem Depot Benefit• Exploit quantity discounts from the supplier• Fluctuations in different warehouses even out, and you gain “statistical economies of scale”• Depot need not to be a physical entity (the point is that goods are allocated after orders completed)• (Maybe a depot with inventory can do even better)• Separate warehouses have little purchasing power• Demand fluctuates for the individual warehouse• It is expensive/ impractical to build a depot• (Demand can vary also in the aggregate)Applicability of modelGood application: Steel for conglomerateQuestionable application: Coca-Cola for 7-ElevenProduction lead times:Long ShortInventory surplus:Holding costs (expensive)Cheap, not to say desirable (up to shelf capacity)Inventory shortfall:Order placed on “backlog” at some penaltyCustomer walks (or buys a substitute)System and Problem Description The Allocation AssumptionPolicy 1: Order up to y every periodPolicy 2: Order up to y every m periodsAllocation Assumption (for every period ordering, normal demand)“Every period, we can find a constant v, such that the total inventory at and in transit to the i th warehouse is:“Every period t, we can make an allocation (at the depot) such that the probability of running out at each warehouse is the same at period t+l”Depot Warehouses Supplier Transportationlead time l(1) 1iilvlµσ+++Example when Allocation Assumption holds (identical warehouses)10108l=0L=1 64SupplierWare-housesDemandPeriod t10-6+5=910-4+3=9?l=0L=1SupplierWarehousesPeriod t+1Example when Allocation Assumption is violated (identical warehouses)10108l=0L=1 90SupplierWare-housesDemandPeriod t10-9+8=910-0+0=10?l=0L=1SupplierWarehousesPeriod t+1The Allocation Assumption holds for high µ/σ and low NProbability of A.A. being true according to experiment presented in paper (my experiment in parenthesis) PercentTheoretical ResultEppen and Schrage derive a good theoretical approximation formula for the probability of A.A. being true.µ/σN½ 1 3/2 2 5/2232.6 (35.9)66.3 (66.3)85.8 (86.5)95.2 (95.5)98.8 (98.8)320.1 (19.8)54.7 (53.2)79.8 (79.0)93.0 (93.5)98.1 (98.2)411.4 (10.0)43.1(41.0)73.3 (72.0)90.1 (90.5)97.3 (97.4)57.6 (4.9)36.5 (30.6)68.6 (65.8)88.3 (88.0)96.5 (96.9)64.6 (2.5)29.9 (22.3)63.2 (60.7)86.4 (85.2)96.1 (95.8)72.8 (1.2)24.5 (15.8)59.1 (53.5)84.1 (83.2)95.5 (95.5)81.6 (0.5)20.4 (11.0)54.3 (46.6)82.0 (80.5)94.6 (95.3)The paper does not explain how “negative demand” should be interpreted. This happens frequently in the lower left corner where my experiments gave different results than those of the paperThe paper does not explain how “negative demand” should be interpreted. This happens frequently in the lower left corner where my experiments gave different results than those of the paperSystem and Problem Description The Allocation AssumptionPolicy 1: Order up to y every periodPolicy 2: Order up to y every m periodsPolicy 1: Order Every Period (fixed ordering costs K = 0)But…Problem: Intuitive answer:How much should we order from the factory every period?What should be the value of y?But is this always possible?If we make the A.A., then yes!We should distribute goods so that total goods at and en route to every factory is “the same”How should we distribute the goods that come in to the depot every period?We should order so that the same total inventory level y is achieved every period. (=order last period’s demand)Eppen and Schrage find an analytical expression for the inventory at each warehouseWe know how much is ordered every period (as a function of y)We know how much is ordered every period (as a function of y)We know how the incoming goods are splitup at the DepotWe know how the incoming goods are splitup at the DepotWe know the (randomfunction for) demand at each warehouse We know the (randomfunction for) demand at each warehouse Eppen and Schrage derive this expression for the inventory S at each warehouse (simplified form for the case of identical warehouses):Random componentFixed component111(1) (1)LNLlittij jttLeySl l eNNµµ+===+=+ + −+ − −∑∑∑The problem is now equivalent to the newsboy problem, and can be solved analytically“The newsboy buys i newspapers, at a cost c each. He sells what is demanded d (random variable), or all he has got i, whichever is less, at a price r. Any surplus is lost.”Newsboy problemCost of surplus (per unit)Cost of shortage (per unit)Deterministic inventoryRandom inventoryNewsboyi -d c r-cOur system (at warehouse) hp111(1) (1)LNLlittijttLeyll eNNµµ+===+++−+− −∑∑∑System and Problem Description The Allocation