1. What is Inventory?
Definition--The stock of any item or resource used in an organization
Raw materials
Finished products
Component parts
Supplies
Work in process
Raw material
- Suppliers may only ship in batches
- Larger purchases many reduces costs
Finished Products
- When it has strategic Importance
- Level capacity plans
- Products displayed to customers (Costco)
Work in Process
- Uncouples phases of production
- Reduced material handling costs
An inventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be.
Supply Chain Management - Aggregate Planning and Inventory Control

2. Inventory System Purpose
The set of policies and controls that determine what inventory levels should be maintained, when stock should be replenished, and how large orders should be
Raw material
- Suppliers may only ship in batches
- Larger purchases many reduces costs
Finished Products
- When it has strategic Importance
- Level capacity plans
- Products displayed to customers (Costco)
Work in Process
- Uncouples phases of production
- Reduced material handling costs
An inventory system is the set of policies and controls that monitor levels of inventory and determines what levels should be maintained, when stock should be replenished, and how large orders should be.
Supply Chain Management - Aggregate Planning and Inventory Control

3. Purposes of Inventory 1. To maintain independence of operations
2. To meet variation in product demand
3. To allow flexibility in production scheduling
4. To provide a safeguard for variation in raw material delivery time
5. To take advantage of economic purchase-order size
1. WIP
2. FGI
3. FGI - relieve pressure caused by customer demand
4. Raw material
5. raw material
Why not hold inventories?
- carrying costs
- lose of responsiveness
- diluted ROI (inventories are an investment)
- Reduced Capacity (inventory eats capacity)
- decreased quality caused by large lot production
Supply Chain Management - Aggregate Planning and Inventory Control

4. Inventory Costs Holding (or carrying) costs
Setup (or production change) costs
Ordering costs
Shortage (or backlog) costs
Holding (or carrying) costs.
Costs for storage, handling, insurance, etc.
Setup (or production change) costs.
Costs for arranging specific equipment setups, etc.
Ordering costs.
Costs of someone placing an order, etc.
Shortage costs.
Costs of canceling an order, etc.
Supply Chain Management - Aggregate Planning and Inventory Control

5. Independent vs. Dependent Demand
Independent Demand
(Demand not related to other items)
Dependent Demand
(Derived/Calculated)
Supply Chain Management - Aggregate Planning and Inventory Control

6. Classifying Inventory Models
Fixed-Order Quantity Models
Event triggered
Make exactly the same amount
Use re-order point to determine timing
Fixed-Time Period Models
Time triggered
Count the number needed to re-order
Fixed -order quantity models - EOQ
Event trigger - level of inventory (order point)
Must constantly monitor inventory
Fixed time period model - periodic review
Order up to
Larger average inventory - covers uncertianty in L+T
Supply Chain Management - Aggregate Planning and Inventory Control

7. Inventory Control Inventory Inventory Models Fixed Order
Quantity Models
Fixed Time
Period Models
Single
Period Models
Constant
Demand
Uncertainty
in Demand
EOQ w/
Quantity
Discounts
Simple
EOQ
EOQ
w/usage
Find the
EOQ and R
Determine
p and d
Calculate
Total costs
Find the
EOQ and R
Select Q
and find R
Supply Chain Management - Aggregate Planning and Inventory Control

8. Fixed-Order Quantity Models Assumptions
Demand for the product is constant and uniform throughout the period
Lead time (time from ordering to receipt) is constant
Price per unit of product is constant
Inventory holding cost is based on average inventory
Ordering or setup costs are constant
All demands for the product will be satisfied (No back orders are allowed)
Define Lead Time
price per unit is constant throughout time period
Supply Chain Management - Aggregate Planning and Inventory Control

9. EOQ Model--Basic Fixed-Order Quantity Model
R = Reorder point
Q = Economic order quantity
L = Lead time L Q R
Time
Inventory
Level
What is average inventory???
Reorder point -
Inventory position --> on hand +on order - backlogs
Supply Chain Management - Aggregate Planning and Inventory Control

10. Basic Fixed-Order Quantity Model
Total Annual Cost =
Annual
Purchase
Cost
Ordering
Holding +
Derive the Total annual Cost Equation, where:
TC - Total annual cost
D - Annual demand (and d-bar = average daily demand = D/365)
C - Cost per unit
Q - Order quantity
S - Cost of placing an order or setup cost
R - Reorder point
L - Lead time
H - Annual holding and storage cost per unit of inventory
TC = DC + (D/Q)S + (Q/2)H
TC/Q = -DQ-2S + (1/2)H
set 0=-DQ-2S = (1/2)H
DQ-2s = (1/2)H
(2DS)/H = Q2
Q = sq. rt. of (2DS)/H
Supply Chain Management - Aggregate Planning and Inventory Control

11. Cost Minimization Goal
Order Quantity (Q) C O S T
Total Cost
Holding
Costs
Annual Cost of
Items (DC)
Ordering Costs
QOPT
Supply Chain Management - Aggregate Planning and Inventory Control

12. average demand per time unit
Deriving the EOQ
Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero
Reorder Point, R = dL d =
average demand per time unit _ L =
Lead time (constant)
Supply Chain Management - Aggregate Planning and Inventory Control

13. EOQ Example Annual Demand (D) = 1,000 units
Days per year considered in average daily demand = 365
Cost to place an order (S) = $10
Holding cost per unit per year (H) = $2.40
Lead time (L) = 7 days
Cost per unit (C) = $15
Determine the economic order quantity and the reorder point.
Supply Chain Management - Aggregate Planning and Inventory Control

14. Solution Why do we round up? 91 or 92 units???
When the inventory level reaches 20, order 91 units.
Supply Chain Management - Aggregate Planning and Inventory Control

15. Problem Retailer of Satellite Dishes How much should we order?
D = 1000 units
S = $ 25
H = $ 100
How much should we order?
What are the Total Annual Stocking Costs?
EOQ = (2DS)/H^.5 = ((2*1000*25)/100)^.5 = 500^.5 = 22.3 or 22
TSC = (1000/22)*25 + (22*100)/2 = = 2236
Supply Chain Management - Aggregate Planning and Inventory Control

16. EOQ with Quantity Discounts
What if we get a price break for buying a larger quantity?
To find the lowest cost order quantity:
Since “C” changes for each price-break, H=iC
Where, i = percentage of unit cost attributed to carrying inventory
and , C = cost (or price) per unit
Find the EOQ at each price break.
Identify relevant and feasible order quantities.
Compare total annual costs
The lowest cost wins.
Supply Chain Management - Aggregate Planning and Inventory Control

17. EOQ with Quantity Discounts Example
Copper may be purchased for
$ .82 per pound for up to 2,499 pounds
$ .81 per pound for 2,500 to 5,000 pounds
$ .80 per pound for orders greater than 5,000 pounds
Demand (D) = 50,000 pounds per year
Holding costs (H) are 20% of the purchase price per unit
Ordering costs (S) = $30
How much should the company order to minimize total costs?
Supply Chain Management - Aggregate Planning and Inventory Control

18. Problem 28 (Costs in $,000) (Order Quantity 100's of units) 44 43
Feasible
(Costs in $,000)
<2500 42
<
>5000 41 40 20 40 60 80
100
(Order Quantity 100's of units)
Supply Chain Management - Aggregate Planning and Inventory Control

19. Inventory Control Inventory Inventory Models Fixed Order
Quantity Models
Fixed Time
Period Models
Single
Period Models
Constant
Demand
Uncertainty
in Demand
Find the L
Find Z
Safety Stock
Find the
EOQ and R
Supply Chain Management - Aggregate Planning and Inventory Control

20. What if demand is not Certain?
Use safety stock to cover uncertainty in demand.
Given: service probability which is the probability demand will NOT exceed some amount.
The safety stock level is set by increasing the reorder point by the amount of safety stock.
The safety stock equals z•L
where,
L = the standard deviation of demand during the lead time.
For example for a 5% chance of running out z 1.65
Assuming demand is N(,2)
Supply Chain Management - Aggregate Planning and Inventory Control

21. Problem Annual Demand = 25,750 or 515/wk @ 50 wks/year
Annual Holding costs = 33% of item cost ($10/unit)
Ordering costs are $250.00
d = 25 per week Leadtime = 1 week
Service Probability = 95%
Find:
a.) the EOQ and R
b.) annual holding costs and annual setup costs
c.) Would you accept a price break of $50 per order for lot sizes that are larger than 2000?
Supply Chain Management - Aggregate Planning and Inventory Control

22. Inventory Control Inventory Inventory Models Fixed Order
Quantity Models
Fixed Time
Period Models
Single
Period Models
Current Inventory
Find theT+L
Find Z
Find order
quantity (q)
Supply Chain Management - Aggregate Planning and Inventory Control

23. Fixed-Time Period Models
Check the inventory every review period and then order a quantity that is large enough to cover demand until the next order will come in.
The model assumes uncertainty in demand with safety stock added to the order quantity.
More exposure to variability than fixed-order models
Current inventory is I
the review period is T
quantity to be ordered is q
The amount of SS is larger than that of the Fixed Order Quantity
Supply Chain Management - Aggregate Planning and Inventory Control

24. Fixed-Time Period Model with Safety Stock Formula
q = Average demand + Safety stock - Inventory currently on hand
q = demand during lead time and review period + SS - on hand
more exposure to variability than fixed order quantity
Supply Chain Management - Aggregate Planning and Inventory Control

25. Determining the Value of sT+L
(number of days exposed to stockout)
The standard deviation of a sequence of random events equals the square root of the sum of the variances.
Supply Chain Management - Aggregate Planning and Inventory Control

26. Example of the Fixed-Time Period Model
Given the information below, how many units should be ordered?
Average daily demand for a product is 20 units.
The review period is 30 days, and lead time is 10 days.
Management has set a policy of satisfying 96 percent
of demand from items in stock. At the beginning of the
review period there are 200 units in inventory. The daily
demand standard deviation is 4 units.
T=30
d=20
L=10
P=0.96
I=200
Supply Chain Management - Aggregate Planning and Inventory Control

27. Example of the Fixed-Time Period Model: Solution
So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period.
Supply Chain Management - Aggregate Planning and Inventory Control

28. Problem A pharmacy orders antibiotics every two weeks (14 days).
the daily demand equals 2000
the daily standard deviation of demand = 800
lead time is 5 days
service level is 99 %
present inventory level is 25,000 units
What is the correct quantity to order to minimize costs?
Supply Chain Management - Aggregate Planning and Inventory Control

29. Inventory Control Inventory Inventory Models Fixed Order
Quantity Models
Fixed Time
Period Models
Single
Period Models
Constant
Demand
Uncertainty
in Demand
Supply Chain Management - Aggregate Planning and Inventory Control

30. Single – Period Model for items w/obsolescence (newsboy problem)
For a single purchase
Amount to order is when marginal profit (MP) is equal to marginal loss (ML).
Adding probabilities
(P = probability of that unit being sold)
for the last unit ordered we want
P(MP)(1-P)ML or P  ML /(MP+ML)
Increase order quantity as long as this holds.
Supply Chain Management - Aggregate Planning and Inventory Control

31. Single–Period Model (Text Prob. #21)
Famous Albert’s Cookie King
Demand
(dozen)
Probability of Demand
1,800
0.05
2,000
0.10
2,200
0.20
2,400
0.30
2,600
2,800
3,000
Each dozen sells for $0.69
and costs $0.49 with a
salvage value of $0.29.
How many cookies
should he bake?
Supply Chain Management - Aggregate Planning and Inventory Control

32. ABC Classification System
Items kept in inventory are not of equal importance in terms of:
dollars invested
profit potential
sales or usage volume
stock-out penalties 60
% of
$ Value A 30 B C
% of
Use 30 60
Inventory Control - Accuracy is recommended to be higher for A items (± .2%) versus ± 1% for B’s and ± 5% for C’s
So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 50% are the “C” items.
Supply Chain Management - Aggregate Planning and Inventory Control

33. Inventory Accuracy and Cycle Counting
Do inventory records agree with physical count?
Cycle Counting
Frequent counts
When? (zero balance, backorder, specified level of activity, level of important item, etc.)
Which Items to count? Most important (ABC analysis)
Items that are at an appropriate time
When to count? Low or zero balance
Backorder with positive inventory
After some activity
By Whom? Designated inventory control people
By the people removing or adding to inventory
Supply Chain Management - Aggregate Planning and Inventory Control