week 6 - inventory theory
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COMM341: Operations Management
Inventory ManagementG. Pond
• Introduction• Single-Period Probabilistic Demand• Multi-Period Fixed Demand• Bulk Purchase Discounts• Safety Stock• Periodic Review• In Practice
Agenda
Real Life Inventory
Inventory Theory aims to answer two basic questions:
1) How much should I order?
2) When should I order it?
Introduction
Objective: Minimize costs
CostsHolding costsSet-up costs Ordering costsBackorder costs
Introduction
1000
0
3000
0
5000
0
7000
0
9000
0
1100
00
1300
00
1500
00
1700
00
1900
00$0
$20$40$60$80
$100$120$140
Annual Holding Costs Annual Ordering Costs
Order Size
Co
st ×
1,0
00
We have different models for:
1) Single period2) Multi-period3) Probabilistic demand4) Fixed demand
Introduction
This is a good model for:• Orders you’ll make only once (e.g., promotional
material for a special event)• Orders related to a discrete event (e.g., an
order you’ll make once annually, batch operations – travel).
Single Period Probabilistic Demand
Single Period Probabilistic Demand
𝑃=𝑐𝑢
𝑐𝑢+𝑐𝑜
If we’re dealing with sufficiently large volumes (say, >30), we can use the following formula to model the probability of failing to sell inventory:
where:
is the cost of underestimating demand is the cost of overestimating demand
Once you’ve found , you can then obtain the corresponding z-value by using a z-table OR by using Excel:
=norm.s.inv()
Single Period Probabilistic Demand
You can then find the corresponding optimum inventory level:
Single Period Probabilistic Demand
Assumptions
• The demand rate is constant - there are no fluctuations in demand. Therefore future demand is known precisely.
• All costs related to holding stock, the unit cost of purchasing new stock, and the cost of placing an order, are all constant.
• As soon as inventory is depleted, an order of new stock arrives.
• The number of units purchased on each stock order is constant - each order size is the same.
Multi-Period Fixed Order Quantity
Multi-Period Fixed Order Quantity
10
Time
Inve
nto
ryAverage Inventory
Level
QQ/2
Multi-Period Fixed Order Quantity
+ Purchasing Cost
Multi-Period Fixed Order Quantity
We can find the minima of this function by using calculus and equating the result to 0
Multi-Period Fixed Order Quantity
1000
0
3000
0
5000
0
7000
0
9000
0
1100
00
1300
00
1500
00
1700
00
1900
00$0
$50
$100
$150
Annual Holding CostsOrder Size
Co
st ×
1,0
00
Now solve for :
We call this the “economic order quantity”
Multi-Period Fixed Order Quantity
Great! That tells me how much to order. But when do I order it?
where:
Reorder Point () - the inventory position (not date or time) at which point new stock should be ordered.
Lead-Time () - the time between when the order is placed and when the ordered stock arrives on site
Average Demand () – is the average demand per day/week/month (just be sure the unit is consistent with lead-time)
Multi-Period Fixed Order Quantity
Example
Suppose the Canadian Forces expends 250,000 rounds of 7.62mm ammunition annually. The average cost of a single round is approximately $0.50 . In consideration of the special safety requirements of storing ammunition, suppose that the holding rate is approximately 85% of the unit cost. Finally, placing an order is estimated to cost approximately $5,000 in labour and shipping charges. What is the order size that minimizes total cost?
Multi-Period Fixed Order Quantity
Multi-Period Deterministic Demand
1000
0
3000
0
5000
0
7000
0
9000
0
1100
00
1300
00
1500
00
1700
00
1900
00$0
$20$40$60$80
$100$120$140
Annual Holding Costs Annual Ordering Costs
Co
st ×
1,0
00
76,697
What do I do when my supplier laughs at me for ordering 76,697 round?
Multi-Period Fixed Order Quantity
Order the closest batch size available (or to be more accurate, compare the total cost of the two nearest batch sizes)
Example
In many cases, price reductions are available for buying in larger quantities. Reconsider the problem of ammunition procurement. Currently, each cartridge purchased costs 50¢. Now imagine that if more than 80,000 rounds are purchased, the price per round is reduced to 45¢ per round, and if more than 90,000 rounds are ordered, the price per round is again reduced to only 40¢ per round.
Bulk Purchase Discounts
Bulk Purchase Discounts
90,000 rounds
Bulk Purchase Discounts
A casino uses 4,000 light bulbs a year. Light bulbs are priced as follows: 1 to 499, 90 cents each; 500 to 999, 85 cents each; and 1,000 or more, 80 cents each. It costs approximately $30 to prepare a purchase order, receive, and pay for it. The holding cost rate is 40% of the purchase price per year. Determine the optimal order quantity and the total annual cost.
Now You Try One!
ExampleConsider the case where the Canadian Forces expects that the lead-time demand for ammunition can be modelled by a normal distribution having a mean of 25,000 rounds during the lead-time period, with a standard deviation of 4,000 rounds. The department is prepared to accept being short-stocked 1% of the time.
Safety Stock
Using Excel,
=norm.s.inv(.99)
yields
Safety Stock
If I accept a stockout 1%, the implication is that I must have sufficient stock the
remaining 99% of the time
Example
Safety Stock
Safety Stock (SS)
Remember that variance is additive (but standard deviation is not).
Safety Stock
Periodic Review
0 5 10 15 20 250
5
10
15
20
25
30
Week
Inve
nto
ry L
evel
Order #1Submitted
Order #1Received
Order #2
Order #2
𝑇 𝐿
Periodic Review
Inventory Level (on-hand + on-order)
Safety-stock
Demand over the lead-time and inter-review period
Evaluating Inventory Policies
Higher is typically better but this is context dependent.
• Accurate Forecasting• Assuming normally distributed variables• Inaccurate inventory data (cycle counting req’d)• Limited Shelf-Life• Safe Storage• Inventory tracking (knowing its location in a
warehouse)
Common Challenges
Cycle Counting• Minimize discrepancies in inventory data• Determine root cause of discrepancy and
correct it.
Typical Solutions
ABC Classification• Devote the majority of your attention to your “Very
Important” (A) stock items
• Give some attention to “Moderately Important” (B) stock items
• Give little attention to the “Least Important” (C) stock items.
Typical Solutions
15-20% of SKUs but 70-80% of annual dollar value
50-60% of SKUs but 5-10% of annual dollar value
Technological Solutions
RFIDs / Transponders
Live Vehicle Tracking
http://tracker.geops.ch/?z=16&s=1&x=-8836519.3105&y=5411201.0721&l=transport
Bar Coding
Technological Solutions
Technological Solutions
Automated Warehouses
Inventory Management Software
How much should I make in a lot/batch?
Other Applications
Figure 5: Graphical representation of the economic lot-size problem.
01
Inv
en
tory
Time
�
� � � �
Production Phase Non-Production
Phase 𝑄∗=√ 2𝐷𝑆
(1− 𝐷𝑃 )𝐻
How far should I allow myself to be short-stocked?
Other Applications
01
Inv
en
tory
Time
� െ��
െ��
Ͳ
� � � �
�
ଵݐ
ଶݐ
𝑆∗=𝑄∗( 𝐻𝐻+𝐶𝑏
)
• Review Chapter 10• Try the following problems from your text:
Problem #6Problem #18Problem #21
• Read Chapter 3 in preparation for next week
Before Next Week
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