INVENTORY MODELINGBasic Concepts

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.

AVERAGE INVENTORYINVENTORY VS. TIME

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

THE EOQ TOTAL COST EQUATIONTC(Q) = CO(D/Q) + Ch(Q/2) + 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

*

*

o

*

*

*

Total Annual Holding and Ordering Costs

Q

TV(Q)

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 ORDER QUANTITY FOR ALLEN

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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