INVENTORY MODELING Basic Concepts

INVENTORY MODELING What is inventory? Items in inventory in a store Manufactured items waiting to be shipped Employees in a firm Computer information in computer files Etc.

COMPONENTS OF AN INVENTORY POLICY Q = the amount to order (the order quantity) R = the number of items left in inventory when an order is placed (the reorder point)

BASIC CONCEPT Balance the cost of having goods in inventory to other costs such as: –Order Cost –Purchase Costs –Shortage Costs

HOLDING COSTS Costs of keeping goods in inventory –Cost of capital –Rent –Utilities –Insurance –Labor –Taxes –Shrinkage, Spoilage, Obsolescence

Holding Cost Rate Annual Holding Cost Per Unit These factors, individually are hard to determine Management (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, C h C h = HC (in $/item in inv./year)

ORDER/SETUP COSTS When purchasing items, this cost is known as the order cost, C O (in $/order) These are costs associated with the ordering process that are independent of the size of the order-- invoice processing, check writing, s, phone calls, accounting etc. – Labor –Communication –Some transportation

ORDER/SETUP COSTS (Cont’d) When these costs are associated with producing items for sale they are called set-up costs (still labeled C O -- in $/setup) Costs associated with getting the process ready for production (regardless of the production quantity) –Readying machines –Calling in shift workers –Paperwork, communications involved

PROCUREMENT/PRODUCTION COSTS These 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 COSTS Shortage/Goodwill Costs associated with being out of stock –goodwill –loss of future sales –labor/communication Fixed administrative costs = C b ($/occurrence) Annualized Customer Waiting Costs = C s ($/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) minimize!! This is a quantity we wish to minimize!!

REVIEW SYSTEMS Continuous Review -- –Items are monitored continuously –When 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 HORIZONS Infinite Time Horizon –Assumes the process has and will continue “forever” Single Period Models –Ordering for a one-time occurence

EOQ-TYPE MODELS EOQ (Economic Order Quantity)-type models assume: Infinite Time Horizon Continuous Review Demand is relatively constant

THE BASIC EOQ MODEL Order the same amount, Q, each time Reordering is instantaneous No shortages –Since reordering is instantaneous Infinite Time Horizon Continuous Review Demand is relatively constant at D items/yr.

AVERAGE INVENTORY INVENTORY VS. TIME QQQ Average Inventory = Q/2 Q/2

THE EOQ COST COMPONENTS Total Annual Order Costs: C O (D/Q) (Cost/order)(average # orders per year) = C O (D/Q) Total Annual Holding Costs: C h (Q/2) (Cost Per Item in inv./yr.)(Average inv.) = C h (Q/2) Total Annual Purchase Costs: CD (Cost Per Item)(Average # items ordered/yr.) = CD

THE EOQ TOTAL COST EQUATION TC(Q) = C O (D/Q) + C h (Q/2) + CD This a function in one unknown (Q) that we wish to minimize

SOLVING FOR Q* TC(Q) = C O (D/Q) + C h (Q/2) + CD

THE REORDER POINT, r* r* = 0Since reordering is instantaneous, r* = 0 L yrs.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 stock r* = LD + SSThen,r* = LD + SS

TOTAL ANNUAL COST The optimal policy is to order Q* when supply reaches r* TC(Q*) = C O D/Q* + (C h /2)(Q*) + CD + C h SS The optimal policy minimizes the total variable cost, hence the total annual cost  Variable Costs TV(Q)  Purchase Costs Safety Stock Costs

TOTAL VARIABLE COST CURVE C H (Q/2) Holding Costs C O D/Q Order Costs C H Q/2 + C O D/Q TOTAL VARIABLE COSTS Q* Optimal Order Quantity occurs where Holding Costs = Reorder Costs Ignoring purchase costs and safety stock costs:

EXAMPLE -- ALLEN APPLIANCE COMPANY Juicer Sales For Past 10 weeks Using 10-period moving average method, 6240/yr D = ( …+ 130)/10 = 120/ wk = 6240/yr

ALLEN APPLIANCE COSTS $10Juicers cost $10 each and sell for $ %Cost of money = 10% 4%Other misc. inventory = 4% $8Labor, postage, telephone/order = $8 $12/hr min.Workers paid $12/hr min. to unload an order 13Desires a safety stock = 13 EOQ Model H.14 H = =.14 C H $1.40 C H =.14(10) = $1.40 C O $12 C O = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12 SS13 SS = 13

OPTIMAL ORDER QUANTITY FOR ALLEN

OPTIMAL QUANTITIES Total Order Cost = C O D/Q* = (12)(6240)/327 = $ Total Holding Cost = (C h /2)Q* = (1.40/2)(327) = $ –(Total Order Cost = Total Holding Cost -- except for roundoff) # Orders Per Year = D/Q* = 6240/327 = Time between orders (Cycle Time) = Q*/D = 327/6240 =.0524 years = 2.72 weeks r* = SS = 13

TOTAL ANNUAL COST Total Variable Cost = Total Order Cost + Total Holding Cost = $ $ = $ Total Purchase Cost = CD = 10(6240) = $62,400 Total Safety Stock Cost =C h SS =(1.40)(13) = $18.20 $62, Total Annual Cost = $ $62,400 + $18.20 = $62,876.09

Using the Inventory Template Input Parameters Note: C h is automatically calculated Optimal Quantities

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

Review Cost Components of Inventory Models –Holding, Order/Setup, Procurement, Shortage Objective -- Minimize Total Annual Cost Continuous Review/Infinite Time Horizon Basic EOQ Assumptions Basic EOQ Formula Reorder Point and Safety Stock Quantities of Interest Use of Template Importance of EOQ Models