week 6 - inventory theory
DESCRIPTION
Week 6 - Inventory TheoryTRANSCRIPT
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COMM341: Operations Management
Inventory ManagementG. Pond
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• Introduction• Single-Period Probabilistic Demand• Multi-Period Fixed Demand• Bulk Purchase Discounts• Safety Stock• Periodic Review• In Practice
Agenda
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Real Life Inventory
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Inventory Theory aims to answer two basic questions:
1) How much should I order?
2) When should I order it?
Introduction
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Objective: Minimize costs
CostsHolding costsSet-up costs Ordering costsBackorder costs
Introduction
1000
0
3000
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5000
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7000
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9000
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1100
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1300
00
1500
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1700
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1900
00$0
$20$40$60$80
$100$120$140
Annual Holding Costs Annual Ordering Costs
Order Size
Co
st ×
1,0
00
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We have different models for:
1) Single period2) Multi-period3) Probabilistic demand4) Fixed demand
Introduction
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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
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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
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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
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You can then find the corresponding optimum inventory level:
Single Period Probabilistic Demand
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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
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Multi-Period Fixed Order Quantity
10
Time
Inve
nto
ryAverage Inventory
Level
QQ/2
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Multi-Period Fixed Order Quantity
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+ Purchasing Cost
Multi-Period Fixed Order Quantity
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We can find the minima of this function by using calculus and equating the result to 0
Multi-Period Fixed Order Quantity
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$50
$100
$150
Annual Holding CostsOrder Size
Co
st ×
1,0
00
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Now solve for :
We call this the “economic order quantity”
Multi-Period Fixed Order Quantity
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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
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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
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Multi-Period Deterministic Demand
1000
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9000
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1100
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1300
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1500
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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
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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)
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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
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Bulk Purchase Discounts
90,000 rounds
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Bulk Purchase Discounts
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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!
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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
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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
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Example
Safety Stock
Safety Stock (SS)
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Remember that variance is additive (but standard deviation is not).
Safety Stock
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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
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Periodic Review
Inventory Level (on-hand + on-order)
Safety-stock
Demand over the lead-time and inter-review period
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Evaluating Inventory Policies
Higher is typically better but this is context dependent.
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• 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
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Cycle Counting• Minimize discrepancies in inventory data• Determine root cause of discrepancy and
correct it.
Typical Solutions
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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
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Technological Solutions
RFIDs / Transponders
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Live Vehicle Tracking
http://tracker.geops.ch/?z=16&s=1&x=-8836519.3105&y=5411201.0721&l=transport
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Bar Coding
Technological Solutions
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Technological Solutions
Automated Warehouses
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Inventory Management Software
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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
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How far should I allow myself to be short-stocked?
Other Applications
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Inv
en
tory
Time
� െ��
െ��
Ͳ
� � � �
�
ଵݐ
ଶݐ
𝑆∗=𝑄∗( 𝐻𝐻+𝐶𝑏
)
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• 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