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

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9000

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1100

00

1300

00

1500

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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

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

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$50

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Annual Holding CostsOrder Size

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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

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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

www.marinetraffic.com

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

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tory

Time

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Ͳ

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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