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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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