Inventory Modeling Concepts Module C4 Inventory Modeling Concepts

What is inventory? INVENTORY MODELING Items in inventory in a store 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 = when to reorder (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, Ch Ch = HC (in $/item in inv./year)

ORDER/SETUP COSTS When 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 writing or checking, phone calls, 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 CO-- 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 = 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 SYSTEMS Continuous Review -- Periodic 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 Single Period Models 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 Demand is relatively constant at D items/yr. Infinite Time Horizon/Continuous Review No shortages Since reordering is instantaneous

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

THE EOQ COST COMPONENTS Total 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 EQUATION TC(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 stock Then, r* = LD + SS

TOTAL ANNUAL COST The optimal policy is to order Q* when supply reaches r* TC(Q*) = COD/Q* + (Ch/2)(Q*) + CD + ChSS <==variable cost==> fixed safety cost stock cost The optimal policy minimizes the total variable cost, hence the total annual cost

TOTAL VARIABLE COST CURVE Ignoring fixed costs and safety stock costs:

EXAMPLE -- ALLEN APPLIANCE COMPANY Juicer Sales For Past 10 weeks 1. 105 6. 120 2. 115 7. 135 3. 125 8. 115 4. 120 9. 110 5. 125 10. 130 Using 10-period moving average method, D = (105 + 115 + …+ 130)/10 = 120/ wk = 6240/yr

This is an EOQ Model with: ALLEN APPLIANCE COSTS Juicers cost $10 each and sell for $11.85 Cost of money = 10% Other misc. costs associated with inventory = 4% Labor, postage, telephone charges/order = $8 Workers paid $12/hr. -- 20 min. to unload an order Desires a safety stock = 13 This is an EOQ Model with: H = .10 + .04 = .14; Ch = .14(10) = $1.40 CO = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12 SS = 13

OPTIMAL ORDER QUANTITY FOR ALLEN

OPTIMAL QUANTITIES Total Order Cost = COD/Q* = (12)(6240)/327 = $228.99 Total Holding Cost = (Ch/2)Q* = (1.40/2)(327) = $228.90 (Total Order Cost = Total Holding Cost -- except for roundoff) # Orders Per Year = D/Q* = 6240/327 = 19.08 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 = $228.99 + $228.90 = $457.89 Total Fixed Cost = CD = 10(6240) = $62,400 Total Safety Stock Cost =ChSS =(1.40)(13) = $18.20 Total Annual Cost = $457.89 + $62,400 + $18.20 = $62,876.09

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

WHY IS 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 fixed costs, only variable costs TV(Q) = COD/Q + (Ch/2)(Q) Now, suppose D really = 7500 (>20% error) We did not know this and got Q* = 327 TV(327) = ((12)(7500))/327 + (1.40/2)(327) =$504.13 Q* should have been: SQRT(2(12)(7500)/1.40) = 359 TV(359) = ((12)(7500))/359 + (1.40/2)(359) =$502.00 This is only a 0.4% increase in the TVCost

Module C4 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 Quantities of Interest Use of Template Importance of EOQ Models