Module C4

Inventory Modeling Concepts

INVENTORY MODELING

What is inventory?• 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 RateAnnual 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 --– 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

• 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 QQ

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

Formula EOQ The 2

2

,

002

*

2

2

h

O

h

O

hO

C

DCQ

C

DCQ

Solving

C

Q

DC

dQ

dTC

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:

• 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

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

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

327 *Q toRound

065.32740.1

)6240)(12(22*

h

O

C

DCQ

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 ParametersNote: Ch is automatically

calculatedOptimal 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 costsTV(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