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**Inventory Modeling Concepts**

Module C4 Inventory Modeling Concepts

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**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.

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**COMPONENTS OF AN INVENTORY POLICY**

Q = the amount to order (the order quantity) R = when to reorder (the reorder point)

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

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**HOLDING COSTS Costs of keeping goods in inventory Cost of capital Rent**

Utilities Insurance Labor Taxes Shrinkage, Spoilage, Obsolescence

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**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)

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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

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**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

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**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

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**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)

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**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!!

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**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

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**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

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**EOQ-TYPE MODELS EOQ (Economic Order Quantity-type models assume:**

Infinite Time Horizon Continuous Review Demand is relatively constant

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**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

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

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**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

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**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

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

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**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

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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

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**TOTAL VARIABLE COST CURVE**

Ignoring fixed costs and safety stock costs:

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**EXAMPLE -- ALLEN APPLIANCE COMPANY**

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

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**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 min. to unload an order Desires a safety stock = 13 This is an EOQ Model with: H = = .14; Ch = .14(10) = $1.40 CO = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12 SS = 13

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**OPTIMAL ORDER QUANTITY FOR ALLEN**

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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 = years = 2.72 weeks r* = SS = 13

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TOTAL ANNUAL COST Total Variable Cost = Total Order Cost + Total Holding Cost = $ $ = $457.89 Total Fixed Cost = CD = 10(6240) = $62,400 Total Safety Stock Cost =ChSS =(1.40)(13) = $18.20 Total Annual Cost = $ $62,400 + $18.20 = $62,876.09

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**Using the Inventory Template**

Input Parameters Note: Ch is automatically calculated Optimal Quantities

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**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

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**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

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**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

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