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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