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Operations Management Inventory Management Chapter 12. Outline. GLOBAL COMPANY PROFILE: AMAZON.COM FUNCTIONS OF INVENTORY Types of Inventory INVENTORY MANAGEMENT ABC Analysis Record Accuracy Cycle Counting Control of Service Inventories . Outline - Continued. INVENTORY MODELS

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    • Types of Inventory
    • ABC Analysis
    • Record Accuracy
    • Cycle Counting
    • Control of Service Inventories
outline continued
Outline - Continued
    • Independent versus Dependent Demand
    • Holding, Ordering, and Setup Costs
    • Basic Economic Order Quantity (EOQ) Model
    • Minimizing Costs
    • Reorder Points
    • Production Order Quantity Model
    • Quantity Discount Models
outline continued1
Outline - Continued
learning objectives
Learning Objectives

When you complete this chapter, you should be able to :

Identify or Define:

  • ABC analysis
  • Record accuracy
  • Cycle counting
  • Independent and dependent demand
  • Holding, Ordering, and Setup Costs

Describe or Explain:

  • The functions of inventory and basic inventory models
amazon com
  • Jeff Bezos, in 1995, started as a “virtual” retailer – no inventory, no warehouses, no overhead; just a bunch of computers.
  • Growth forced to excel in inventory management!
  • AMAZON is now a worldwide leader in warehouse management and automation.
order fulfillment at amazon
Order Fulfillment at AMAZON
  • You order items;, computer assigns your order to distribution center [closest facility that has the product(s)]
  • Lights indicate products ordered to workers who retrieve product and reset light.
  • Items placed in crate with items from other orders, and crate is placed on conveyor. Bar code on item is scanned 15 times – virtually eliminating error.
order fulfillment at amazon continued
Order Fulfillment at AMAZON- Continued
  • Crates arrive at central point where items are boxed and labeled with new bar code.
  • Gift wrapping done by hand (30 packages per hour)
  • Box is packed, taped, weighed and labeled before leaving warehouse in a truck.
  • Order appears on your doorstep within a week
what is inventory
What is Inventory?
  • Stock of materials
  • Stored capacity
  • Examples

© 1995 Corel Corp.

© 1984-1994 T/Maker Co.

© 1984-1994 T/Maker Co.

© 1995 Corel Corp.

the functions of inventory
The Functions of Inventory
  • To ”decouple” or separate various parts of the production process
  • To provide a stock of goods that will provide a “selection” for customers
  • To take advantage of quantity discounts
  • To hedge against inflation and upward price changes
types of inventory
Types of Inventory
  • Raw material
  • Work-in-progress
  • Maintenance/repair/operating supply
  • Finished goods
the material flow cycle
The Material Flow Cycle

Note the proportion of time material spends as inventory as opposed to being actually worked on.

Effective inventory management and materials movement can reduce overall cycle time significantly!

disadvantages of inventory
Disadvantages of Inventory
  • Higher costs
    • Item cost (if purchased)
    • Ordering (or setup) cost
      • Costs of forms, clerks’ wages etc.
    • Holding (or carrying) cost
      • Building lease, insurance, taxes etc.
  • Difficult to control
  • Hides production problems
abc analysis
ABC Analysis
  • Divides on-hand inventory into 3 classes
    • A class, B class, C class
  • Basis is usually annual $ volume
    • $ volume = Annual demand x Unit cost
  • Policies based on ABC analysis
    • Develop class A suppliers more
    • Give tighter physical control of A items
    • Forecast A items more carefully
classifying items as abc

% Annual $ Usage


% $ Vol

% Items






















Classifying Items as ABC

% of Inventory Items

cycle counting
Cycle Counting
  • Physically counting a sample of total inventory on a regular basis
  • Used often with ABC classification
    • A items counted most often (e.g., daily)
advantages of cycle counting
Advantages of Cycle Counting
  • Eliminates shutdown and interruption of production necessary for annual physical inventories
  • Eliminates annual inventory adjustments
  • Provides trained personnel to audit the accuracy of inventory
  • Allows the cause of errors to be identified and remedial action to be taken
  • Maintains accurate inventory records
why do we control inventory
Why do we control inventory?
  • Inventories represent a vast segment of total economic activity.
  • Even minor improvements can create large savings.

How do we control inventory?

  • Application of optimization techniques
  • Information processing and retrieval techniques
decisions of an inventory policy
Decisions of an inventory policy
  • If there is no production, i.e., pure inventory system
    • How much to order? Order quantity
    • When to order? Reorder quantity

Ex:Order Q=100 units when the inventory level drops to ROP=15 units.

  • If there is also production
    • When to start/stop production?
elements of inventory decisions
Elements of Inventory Decisions
  • Costs:
    • Ordering and Procurement costs (or Setup costs in manufacturing systems)
    • Inventory holding or carrying costs
    • Inventory shortage costs
  • Demand structure
    • How does it vary? Certain, uncertain?
  • Supply structure
    • Any capacity limitations, defectives, number of suppliers?
  • Lead times:
    • Certain, uncertain?
ordering and procurement costs
Ordering and Procurement Costs
  • Represent all expenses incurred in ordering or manufacturing items related to
    • Acquisition
    • Transportation
    • Collecting, sorting, placing the items in the storage
    • Managerial and clerical costs associated with order placement.
  • Ordering costs are fixed, independent of the order size.
    • Supplies
    • Forms
    • Order processing
    • Clerical support
  • Procurement costsdepend on the order size.
set up costs
Set-up Costs

In a manufacturing system order costs are realized in the form of “set-up” costs. These include:

  • Clean-up costs
  • Re-tooling costs
  • Adjustment costs
inventory holding or carrying costs
Inventory Holding or Carrying Costs
  • Expenses incurred during the storage of items.
    • Physical Costs: Warehouse operation costs, insurence, property taxes.
    • Pilferage, spoilage, obsolescence
    • Opportunity cost of investing in inventory rather than investing somewhere else, ex. in a bank.
  • Inventory costs are variable costs that depend on the order size.
inventory holding costs approximate ranges

Housing costs (building rent, depreciation, operating cost, taxes, insurance)

Material handling costs (equipment, lease or depreciation, power, operating cost)

Labor cost from extra handling

Investment costs (borrowing costs, taxes, and insurance on inventory)

Pilferage, scrap, and obsolescence

Overall carrying cost

Cost as a

% of Inventory Value


(3 - 10%)


(1 - 3.5%)


(3 -5%)


(6 - 24%)


(2 - 5%)


Inventory Holding Costs(Approximate Ranges)
shortage costs
Shortage Costs
  • Occur whenever the demand is not satisfied. Order is either “backordered” or “lost”.
  • Backordering Costs:
    • Fixed cost of extra managerial work.
    • Loss of customer goodwill: Variable cost that depends on duration of backorder.
  • Lost Sales Costs:
    • Marginal profit that the item would have earned.
    • Loss of customer goodwill.
demand structure
Demand Structure
  • Continuous versus discrete demand

Ex: Natural gas consumption in houses

Detergent consumption in houses

  • Deterministic (certain) versus stochastic (uncertain) demand

Ex: Order quantities for the next months are 20,30,10,50.

Order quantities in a month are normally distributed with mean 25 and variance 4.

  • Constant versus dynamic demand

Ex: Demand quantities for the next months are 20, 21, 20, 19

Demand quantities for the next months are 20, 50, 10, 2

supply structure
Supply Structure
  • Any defectives?

If the received lot includes defective items this brings uncertainty

  • Any capacity limitations?

Do we fully receive what we order?

  • Number of suppliers, fixed or variable?
lead time
Lead time
  • Time elapsed between the order delivery and order receipt.
  • Can be constant or stochastic.

Ex: Lead time is 10 days.

Lead time is between 8-12 days.

inventory models
Inventory Models

Help answer the inventory planning questions!

  • Fixed order-quantity models
    • Economic order quantity
    • Production order quantity
    • Quantity discount
  • Probabilistic models
  • Fixed order-period models

© 1984-1994 T/Maker Co.

the economic order quantity eoq model
The Economic Order QuantityEOQ-Model
  • Decision variable:Q = Order Quantity
  • Parameters:

S = Fixed cost per order ($/order)

D = Annual number of items demanded (unit/year)

P= Unit cost of procuring an item ($/unit)

I= Annual cost of holding a dollar in inventory ($/$/year)

H= Annual cost of holding a unit item in inventory ($/unit/year)


  • Objective is to “minimize total annual cost”.
eoq inventory policy
EOQ Inventory Policy

Average Inv. Level

assumptions of classical eoq model
Assumptions of Classical EOQ Model
  • Demand rate is constant or stable.
  • There is infinite supply availability.
  • Lead time is constant or zero.
  • No quantity discounts are made.
  • All demand is met on time, no backordering, no stockout.
  • Only order (or setup) cost and holding cost
eoq model how much to order

Annual Cost

Total Cost Curve

Holding Cost Curve

Minimum total cost

Order (Setup) Cost Curve

Order quantity

Optimal Order Quantity (Q*)

EOQ ModelHow Much to Order?
costs of eoq model
Costs of EOQ Model
  • Total ordering cost is the number of orders times the cost per order:
  • Total holding cost is the cost per item held 1year times the average inventory:
  • The annual procurement cost is the product of annual demand and unit cost:

Procurement cost = PD

annual cost of eoq model
Annual Cost of EOQ-Model
  • Here PDis not a relevant cost and thus dropped.
  • Minimize Total Annual Inventory Cost:
optimal solution of eoq
Optimal Solution of EOQ
  • Optimal solution is theeconomic order quantity
  • Optimal Total Cost
example the house of wines and liquors
Example:The House of Wines and Liquors

Allex Mullen decides that the first task in utilizing inventory models is to determine the value of model parameters:

  • Annual demand 5200 cases of beer
  • $10 telephone charge for ordering
  • Purchase cost is $1.5/case beer+shipping cost $0.5/case
  • 10%bank interest, 5%state franchise tax, 5% theft insurance rate

How many should he order, how often, and at what annual relevant inventory cost?


The economic order quantity is

  • The inventory cycle duration is

T = Q/D= 510/5200 = 0.098 year or 36 days

  • The total annual relevant inventory cost is:
robustness of eoq model
Robustness of EOQ Model
  • EOQ is a robust model with respect to the estimation errors in D, S, P or I.
  • Let Dactual=4 Destimated

Then EOQactual=2 Destimated


ex the house of wines and liquors
Ex: The House of Wines and Liquors
  • Alex Mullen applies EOQ to another product, a particular variety of Chilean wine that sells 1000 cases annually. The cost is $20 per case. A telephone call to Chile to place an order costs $100. The holding costs are the same as for Tres Equis Beer.
ex the house of wines and liquors1
Ex: The House of Wines and Liquors

T = Q/D= 24/1000 = .224 year or 82 days

the reorder point rop when to give orders


Slope = units/day = d

Inventory level (units)

ROP (Units)

Time (days)

Lead time = LT

The Reorder Point ROPWhen to give orders?



Optimal Inventory Policywith Backordering

Orders placed during shortages are backordered.


Optimal Inventory Policywith Backordering

Imax: Quantity on hand when a shipment arrives.

C:Cost of being one item short for a year

Optimal order quantity and order level

example the house of wines and liquors backorders
Example:The House of Wines and Liquors-Backorders

The marketing department tells Alex that beer is a convenience product that can not be backordered, so sale is lost! However some wine customers are connoisseurs who are willing to order out-of-stock items. Nevertheless, the store owner will incur some penalty cost if there is a shortage of wine.

Suppose that retailer suffers lost profit on future business equal to $0.01/unit each day that a wine is on backorder. What should be the optimal ordering policy if backordering is allowed?

Solution: The order quantity is computed:

p = $.01×365 = $3.65/unit/year.


Example: Solution

  • The max. order level Imax is
  • The relevant cost is

smaller than before, why?

is backordering better
Is backordering better?
  • Fewer orders are placed when there is backordering.
  • Average inventory level is smaller.

Backorders/cycle=Q* – Imax* =324 – 154 = 170 units/cycle.

Proportion of demand not satisfied on time

=(Q*- Imax* )/Q*=170/324= 52.5%Service level = 1- Proportion of demand not satisfied on time = 1-52.5%=47.5%

  • The results suggest that:

Retailers will run short in each cycle.

But can they get away with it?

  • So backordering must make sense!
shortage penalty considerations
Shortage Penalty Considerations

Shortage Cost

Shortage Cost





1 year

1 year

Theoretical Assumption

Case excluded by the model

imputed shortage penalty
Imputed Shortage Penalty

An alternative approach for establishing an inventory policy is based on achieving a desired service level.

Service Level, L is the proportion of demand met on time

Imputed shortage penalty

as c increases eoq is more robust
As C increases EOQ is more robust

D=1000 units/yr S=$100/order P=$20/unit I= $0.20/$/year






EOQ with no









imputed shortage penalty

production order quantity model
Production Order Quantity Model
  • Answers how much to order and when to order
  • Allows partial receipt of material
    • Other EOQ assumptions apply
  • Suited for production environment
    • Material produced, used immediately
    • Provides production lot size
  • Lower holding cost than EOQ model
reasons for variability in production
Reasons for Variability in Production
  • Most variability is caused by waste or by poor management. Specific causes include:
    • employees, machines, and suppliers produce units that do not conform to standards, are late or are not the proper quantity
    • inaccurate engineering drawings or specifications
    • production personnel try to produce before drawings or specifications are complete
    • customer demands are unknown
economic production quantity model
Economic Production-Quantity Model

The EOQ model may be extended to find the optimal production quantity,Q.p: Daily production rate, d : Daily demand rate




poq model equations




POQ Model Equations






Optimal Order Quantity












Maximum inventory level




D = Demand per year

S = Setup cost

H = Holding cost

d = Demand per day

p = Production per day





Setup Cost








0.5 * H * Q

Holding Cost


quantity discount model
Quantity Discount Model
  • Answers how much to order & when to order
  • Allows quantity discounts
    • Reduced price when item is purchased in larger quantities
    • Other EOQ assumptions apply
  • Trade-off is between lower price & increased holding cost
probabilistic models
Probabilistic Models
  • Answer how much & when to order
  • Allow demand to vary
    • Follows normal distribution
    • Other EOQ assumptions apply
  • Consider service level, L & safety stock
    • Service level = L =1 - Probability of stockout
    • Higher service level means more safety stock
      • More safety stock means higher ROP
probabilistic models when to order

Service Level


Inventory Level



Order Quantity




Reorder Point (ROP)

Safety Stock (SS)

Place order

Receive order


Lead Time

Probabilistic ModelsWhen to Order?
fixed period model
Fixed Period Model
  • Answers how much to order
  • Orders placed at fixed intervals
    • Inventory brought up to target amount
    • Amount ordered varies
  • No continuous inventory count
    • Possibility of stockout between intervals
  • Useful when vendors visit routinely
    • Example: P&G representative calls every 2 weeks
inventory level in a fixed period system

Target maximum





On-Hand Inventory





Inventory Level in a Fixed Period System

Various amounts (Qi) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximum

fixed period model when to order

Inventory Level

Target maximum





Fixed Period ModelWhen to Order?