INVENTORY MODELINGINVENTORY MODELING

Basic Concepts

INVENTORY MODELINGINVENTORY 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 COMPONENTS OF AN INVENTORY POLICYINVENTORY 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 CONCEPTBASIC CONCEPT

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

HOLDING COSTSHOLDING COSTS

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

Holding Cost RateHolding Cost RateAnnual Holding Cost Per UnitAnnual 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 COSTSORDER/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 processing, check writing, e-mails, phone calls, accounting etc.– Labor– Communication – Some transportation

ORDER/SETUP COSTS (Cont’d)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 PROCUREMENT/PRODUCTION COSTSCOSTS

• 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 CUSTOMER SATISFACTION COSTSCOSTS

• 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 EQUATIONBASIC 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!!minimize!!

REVIEW SYSTEMSREVIEW 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 HORIZONSTIME HORIZONS

• Infinite Time Horizon– Assumes the process has and will continue

“forever”

• Single Period Models – Ordering for a one-time occurence

EOQ-TYPE MODELSEOQ-TYPE MODELS

EOQ (Economic Order Quantity)-type models assume:

• Infinite Time Horizon

• Continuous Review

• Demand is relatively constant

THE BASIC EOQ MODELTHE 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 INVENTORYAVERAGE INVENTORY

INVENTORY VS. TIME

Q QQ

Average Inventory =Average Inventory = Q/2Q/2

Q/2Q/2

THE EOQ COST COMPONENTSTHE EOQ COST COMPONENTS

• Total Annual Order Costs:(Cost/order)(average # orders per year) = CCOO(D/Q)(D/Q)

• Total Annual Holding Costs:(Cost Per Item in inv./yr.)(Average inv.) = CChh(Q/2)(Q/2)

• Total Annual Purchase Costs: (Cost Per Item)(Average # items ordered/yr.) = CDCD

THE EOQ TOTAL COST THE EOQ TOTAL COST EQUATIONEQUATION

• TC(Q) = CO(D/Q) + Ch(Q/2) + CD

• This a function in one unknown (Q) that we wish to minimize

CDQ

CQ

DCQTC ho

2)(

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

Formula EOQThe 2

2

,

002

*

2

2

h

O

h

O

hO

C

DCQ

C

DCQ

Solving

C

Q

DC

dQ

dTC

CDQC

DQCQTC ho

2)( 1

THE REORDER POINT, r*THE REORDER POINT, r*

• Since reordering is instantaneous, r* = 0r* = 0

• MODIFICATION -- fixed lead time = L yrs.L yrs.

r* = LDr* = 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 + SSr* = LD + SS

TOTAL ANNUAL COSTTOTAL ANNUAL COST• The optimal policy is to order Q* when

supply reaches r*

TC(Q*) = COD/Q* + (Ch/2)(Q*) + CD + ChSS

• The optimal policy minimizes the total variable cost, hence the total annual cost

Variable Costs TV(Q) PurchaseCosts

SafetyStockCosts

TOTAL VARIABLE COST CURVETOTAL VARIABLE COST CURVE

• The Total Variable Costs function

Constructing the Total Annual Variable Cost CurveAdd the two curves to one another

* * o * * *Total Annual Holding and Ordering Costs

Q

TV(Q)

Q*

The optimal order size

CH(Q/2)

Holding

Costs

COD/Q

Order Costs

CHQ/2 + COD/QTOTAL VARIABLE COSTS

Q*Q* Optimal Order Quantity occurs where Holding Costs = Reorder Costs

Ignoring purchase costs and safety stock costs:

EXAMPLE -- ALLEN EXAMPLE -- ALLEN APPLIANCE COMPANYAPPLIANCE 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/yr6240/yr

ALLEN APPLIANCE COSTSALLEN APPLIANCE COSTS• Juicers cost $10$10 each and sell for $11.85

• Cost of money = 10%10%

• Other misc. inventory = 4%4%

• Labor, postage, telephone/order = $8$8• Workers paid $12/hr.--$12/hr.--20 min.20 min. to unload an order

• Desires a safety stock = 1313

EOQ ModelEOQ Model

HH = .10 + .04 = .14.14CCHH = .14(10) = $1.40$1.40

CCOO = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12$12

SSSS = 1313

OPTIMAL ORDER QUANTITY OPTIMAL ORDER QUANTITY FOR ALLENFOR ALLEN

327 *Q toRound

065.32740.1

)6240)(12(22*

h

O

C

DCQ

OPTIMAL QUANTITIESOPTIMAL 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 COSTTOTAL ANNUAL COST

• Total Variable Cost = Total Order Cost + Total Holding Cost = $228.99 + $228.90 = $457.89

• Total Purchase 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$62,876.09

Using the Inventory TemplateUsing the Inventory Template

Input ParametersNote: Ch is automatically

calculatedOptimal Quantities

WHY IS THE EOQ MODEL WHY IS THE EOQ MODEL IMPORTANT?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 MODELSINSENSIVITY IN EOQ MODELS• We cannot affect purchase costs and safety stock cost,

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* = 327Q* = 327TV(327)TV(327) = ((12)(7500))/327 + (1.40/2)(327) = $504.13$504.13

QQ** should have been: SQRT(2(12)(7500)/1.40) = 359359

TV(359)TV(359) = ((12)(7500))/359 + (1.40/2)(359) = $502.00$502.00

• This is only a 0.4%0.4% increase in the TVCost

ReviewReview• 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