INVENTORY -- Basic Concepts

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  • INVENTORY MODELINGWhat is inventory?Items in inventory in a storeManufactured items waiting to be shippedEmployees in a firmComputer information in computer filesEtc.

  • COMPONENTS OF AN INVENTORY POLICYQ = the amount to order (the order quantity)

    R = the number of items left in inventory when an order is placed (the reorder point)

  • BASIC CONCEPTBalance the cost of having goods in inventory (Holding Cost) to other costs such as:

    Order CostPurchase CostsShortage Costs

  • HOLDING COSTSCosts of keeping goods in inventoryCost of capitalRentUtilitiesInsuranceLaborTaxesShrinkage, Spoilage, Obsolescence

  • Holding Cost RateAnnual Holding Cost Per UnitThese factors, individually are hard to determineManagement (typically the CFO) assigns a holding cost rate, H, which is a percentage of the value of the item, C

    Annual Holding Cost Per Unit, Ch

    Ch = HC (in $/item in inv./year)

  • ORDER/SETUP COSTSWhen purchasing items, this cost is known as the order cost, CO (in $/order) These are costs associated with the ordering process that are independent of the size of the order-- invoice processing, check writing, e-mails, phone calls, accounting etc. LaborCommunication Some transportation

  • ORDER/SETUP COSTS (Contd)When these costs are associated with producing items for sale they are called set-up costs (still labeled CO-- in $/setup)Costs associated with getting the process ready for production (regardless of the production quantity)Readying machinesCalling in shift workersPaperwork, communications involved

  • PROCUREMENT/PRODUCTION COSTSThese are the per unit purchase costs, C, if we are ordering the items from a supplier

    These are the per unit production costs, C, if we are producing the items for sale

  • CUSTOMER SATISFACTION COSTSShortage/Goodwill Costs associated with being out of stockgoodwillloss of future saleslabor/communication Fixed administrative costs = Cb ($/occurrence)Annualized Customer Waiting Costs =Cs ($/item short/year)

  • BASIC INVENTORY EQUATION(Total Annual Inventory Costs) =(Total Annual Order/Setup-Up Costs) +(Total Annual Holding Costs) +(Total Annual Purchase/Production Costs) +(Total Annual Shortage/Goodwill Costs)This is a quantity we wish to minimize!!

  • REVIEW SYSTEMSContinuous Review --Items are monitored continuouslyWhen inventory reaches some critical level, R, an order is placed for additional items

    Periodic Review --Ordering is done periodically (every day, week, 2 weeks, etc.)Inventory is checked just prior to ordering to determine an order quantity

  • TIME HORIZONSInfinite Time HorizonAssumes the process has and will continue forever

    Single Period Models Ordering for a one-time occurence

  • EOQ-TYPE MODELSEOQ (Economic Order Quantity)-type models assume:

    Infinite Time Horizon

    Continuous Review

    Demand is relatively constant

  • THE BASIC EOQ MODELOrder the same amount, Q, each timeReordering is instantaneousNo shortages Since reordering is instantaneousInfinite Time HorizonContinuous ReviewDemand is relatively constant at D items/yr.


  • THE EOQ COST COMPONENTSTotal Annual Order Costs:(Cost/order)(average # orders per year) = CO(D/Q)

    Total Annual Holding Costs:(Cost Per Item in inv./yr.)(Average inv.) = Ch(Q/2)

    Total Annual Purchase Costs: (Cost Per Item)(Average # items ordered/yr.) = CD


    This a function in one unknown (Q) that we wish to minimize

  • SOLVING FOR Q*TC(Q) = CO(D/Q) + Ch(Q/2) + CD

  • THE REORDER POINT, r*Since reordering is instantaneous, r* = 0

    MODIFICATION -- fixed lead time = L yrs.r* = LD But demand was only approximately constant so we may wish to carry some safety stock (SS) to lessen the likelihood of running out of stockThen,r* = LD + SS

  • TOTAL ANNUAL COSTThe optimal policy is to order Q* when supply reaches r*

    TC(Q*) = COD/Q* + Ch (Q*/2) + CD + ChSS

    The optimal policy minimizes the total variable cost, hence the total annual cost Variable Costs TV(Q)Purchase CostsSafety Stock Costs

  • TOTAL VARIABLE COST CURVEQ*Optimal Order Quantity occurs where Holding Costs = Reorder CostsIgnoring purchase costs and safety stock costs:

  • The Total Variable Costs function

    Constructing the Total Annual Variable Cost Curve

    Add the two curves to one another







    Total Annual Holding and Ordering Costs



  • EXAMPLE -- ALLEN APPLIANCE COMPANYJuicer Sales For Past 10 weeks1.1056.1202.1157.1353.1258.1154.1209.1105.12510.130

    Using 10-period moving average method, D = (105 + 115 + + 130)/10 = 120/ wk = 6240/yr

  • ALLEN APPLIANCE COSTSJuicers cost $10 each and sell for $11.85Cost of money = 10% Other misc. inventory = 4%Labor, postage, telephone/order = $8Workers paid $12/hr.--20 min. to unload an orderDesires a safety stock = 13 EOQ ModelD = 6240H = .10 + .04 = .14CH = .14(10) = $1.40CO = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12SS = 13


  • OPTIMAL QUANTITIESTotal Order Cost = CO(D/Q*) = (12)(6240)/327 = $228.99Total Holding Cost = Ch(Q*/2) = (1.40/2)(327) = $228.90(Total Order Cost = Total Holding Cost -- except for rounding error: actual Q* = 327.065)# Orders Per Year = D/Q* = 6240/327 = 19.08Time between orders (Cycle Time) = Q*/D = 327/6240 = .0524 years = 2.72 weeksr* = SS = 13

  • TOTAL ANNUAL COSTTotal Variable Cost = Total Order Cost + Total Holding Cost = $228.99 + $228.90 = $457.89Total Purchase Cost = CD = 10(6240) = $62,400Total Safety Stock Cost =ChSS =(1.40)(13) = $18.20Total Annual Cost = $457.89 + $62,400 + $18.20 = $62,876.09

  • Using the Inventory Template

  • WHY IS THE EOQ MODEL IMPORTANT?No real-life model really is an EOQ model

    Many models are variants of EOQ-type models

    Many situations can be approximated by EOQ models

    The EOQ model is relatively insensitive to some pretty major errors in input parameters

  • INSENSIVITY IN EOQ MODELSWe cannot affect purchase costs and safety stock cost, only variable costs: TV(Q) = COD/Q + Ch(Q/2) Now, suppose D really = 7500 (>20% error)We did not know this and got Q* = 327TV(327) = ((12)(7500))/327 + (1.40/2)(327) = $504.13Q* should have been: SQRT(2(12)(7500)/1.40) = 359TV(359) = ((12)(7500))/359 + (1.40/2)(359) = $502.00This is only a 0.4% increase in the TVCost

  • ReviewCost Components of Inventory ModelsHolding, Order/Setup, Procurement, ShortageObjective -- Minimize Total Annual CostContinuous Review/Infinite Time HorizonBasic EOQ AssumptionsBasic EOQ FormulaReorder Point and Safety StockQuantities of InterestUse of TemplateImportance of EOQ Models

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