Inventory Management Henry C. Co Technology and Operations Management, California Polytechnic and State University

Inventory Management

Henry C. CoTechnology and Operations Management, California Polytechnic and State University

Inventory Management (Henry C. Co) 2

Short-range decisions about supplies, inventories, production levels, staffing patterns, schedules, and distribution -operations infrastructure

4 R’s Right Material Right Amount Right Place Right Time.

Motivations

Economies of ScaleUncertainties


Economies of Scale Over-investment

Ties up capacity and financial resources Inventory carrying cost, obsolescence, etc.

Insufficient/late availability causes Idle personnel or equipment when

components ran out Lost sales/customer-goodwill if items are

out-of-stock.


Uncertainties Safety stock (demand or supply

uncertainty) In-transit inventories (lead times) Hedge inventories

Drivers


Approximately 16 % of total assets are invested in inventories (1986) In manufacturing firms, 25 to 35% of total assets

of typical are tied in inventories Materials’ average share in a manufacturer’s

cost of goods sold 40% in 1945 50% in 1960 > 60% Today

Spare parts (service parts) inventories of a typical manufacturer ~ $ 5 million - $ 15 million


Consequently … distribution and inventory (logistics) costs are quite substantial

Value of inventories in U.S. ~ $ 1 trillion (1993)


Basic definitions An inventory is an accumulation of a

commodity that will be used to satisfy some future demand.

Inventories of the following form: Raw material Components Semi-finished goods Spare parts Purchased products in retailing.


Functional classification Cycle Inventories: Produce or buy in

larger quantities than needed. Economies of scale Quantity discounts Restrictions (technological, transportation,

…)

inventory

time

cycle stock

a cycle


Safety Stock: Provides protection against irregularities and uncertainties.

place order at this time

time

inventory

Safety stockreorder level


Anticipation stock: Low demand in one part of the year build up stock for the high demand season

Low demand season

AverageAnticipation stock


Hedge inventories : expect changes in the conditions (price, strike, supply, etc.)

Pipeline (or work-in-process) inventories: goods in transit, between levels of a supply chain, between work stations.


Service Operations No tangible items to purchase or

inventory, materials management is of minor concern

Operations that provide repair or refurbishment services carry inventory of replacement parts and supplies

Examples: Automobile service centers carry

automotive parts Hospitals carry inventory of food, linens,

medicine, and medical supplies


Functions of Inventory To meet anticipated demand To smooth production requirements To decouple components of the

production-distribution To protect against stock-outs To take advantage of order cycles To help hedge against price increases

or to take advantage of quantity discounts


Conflicting Needs Some Excuses for Holding Excess

Inventory “Inaccurate sales forecast” “Poor quality” “Unsynchronized processes” “Poor schedules” “Unreliable suppliers” “Unreliable shippers” “Poor attitudes”


Pressures to Cut Inventory Interest/opportunity cost Storage and handling Property taxes Insurance premiums Shrinkage INVENTORIES HIDE PROBLEMS!


Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Profile of Inventory Level Over Time


Profile of … Frequent Orders

The Classic EOQ Model


The Economic Order Quantity (EOQ) The total cost curve reaches its minimum

where the carrying and ordering costs are equal.

EOQ represents trade-off between fixed cost associated with production or procurement against inventory holding costs.D= Rate of demand, units/yearS = Fixed cost of procurement, $/orderv = Variable cost of procurement.h = Cost/unit time of holding each unit of

inventory, $/unit/yearQ= Quantity ordered, units


The Classical EOQ Model The total cost curve reaches its

minimum where the carrying and ordering costs are equal.

QO

Ann

ual C

ost

TCQH

D

QS

2

Ordering Costs

Order Quantity (Q)(optimal order quantity)


Using calculus, we take the derivative of the total cost function (TC) and set the derivative (slope) equal to zero and solve for Q.

Cost Holding Annual

Cost) Setupor der Demand)(Or 2(Annual =

H

2DS = QOPT


A grocery store pays $100 for each delivery of milk from the dairy. They sell 200 gallons per week. Each gallon costs the store $2, and they sell it for $3. They earn a 15% return on cash that is invested in milk. They have a large refrigerator that holds up to 2,000 gallons. It costs them $10,000 per year to maintain.

D ~ 200(52) = 10,400 gallons per year.h ~ 15%($2) = $.30 per gallon per yearS ~ $100

Q* = 2,633 Roughly one replenishment every 9 weeks?!?!

Example


Suppose that the refrigerator held only 100 gallons. How much would it be worth to expand capacity to 200 gallons?

415,10$2

1003$.

100

400,10100$)100( TC

230,5$2

2003$.

200

400,10100$)200( TC

$5,185 per year in operational savings from doublingfreezer size.


A bank has determined that it costs $30 to replenish the cash in one of its suburban ATMs. Customers take cash out of the ATM at a rate of $2000 per day (365 days per year). The bank earns a 10% return on cash that is not sitting in an ATM. Let us define $1000 to be a basic unit of cash.

D ~ $2K(365) = $730K per year.h ~ 10%($1K) = $100 per $K per yearS ~ $30

Q* = $20.9K Roughly one replenishment every 10 days.

Example


Potential Analytical Errors Using different time units for holding

cost versus the rate of demand. Determining the true opportunity cost

associated with holding inventory. (Physical costs as well as financial.)

Dealing with operational constraints.


When to Reorder? Reorder Point R – When the quantity

on hand of an item drops to this amount, the item is reordered

Safety Stock SS – Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

Service Level - Probability that demand will not exceed supply during lead time.


Continuous Review

Lead Time Demand

(U = D * LT )

Inventory

time

Reorder Quantity

(Q)

slope = -D

LT

Reorder Point(R)

Cycle Stock

} Safety StockSS

SSUR SSUR

U = Average Lead Time Demand

SS = Safety Stock; R = + SSU


LT

Expected demandduring lead time

Maximum probable demand during lead time

Reorder point R

Qu

ant

ity

Safety stock


Reorder Point

R

Risk of a stock-out

Service level =

probability of no stock-out

Expecteddemand

Safety-stock

0 z

Quantity

z-scale

U


Determining Reorder Point R The Average lead time demand is equal to

the average rate of demand (D) multiplied by the length of the lead time (L).

The amount of safety stock is influenced by the variability of demand, and our preference for avoiding inventory versus satisfying all of the demand.

If distribution of demand is stable over time, and the demand in one interval of time is statistically independent from that in another interval, then the standard deviation of lead time demand is proportional to the square root of the length (L) of the lead time:

DLU

σLσL


Standard Deviation of Lead-time Demand If daily demand has mean 100, and standard

deviation 40, and the lead time L= 9 days: Then Lead Time Demand has mean = 900, and the standard deviation of lead time demand =120 (=SQRT[9] * 40).

If annual demand has mean 1040, and standard deviation 600, and the lead time L = 2 weeks: Then Lead Time Demand has mean = 40 (= 1040*(2/52) and the standard deviation of lead time demand = 117.67 (=SQRT[2/52]*600.


If weekly demand has mean 20, and standard deviation 83.2, and the lead time L = 2 weeks: Then Lead Time Demand has mean = 40 and the standard deviation of lead time demand = 117.67. (Note that this example is identical to the previous one, but the demand was specified in terms of weekly demand instead of annual.)


Safety Stock Safety stock determines the expected

amount of unsatisfied demand Whenever the amount of demand

during the lead time exceeds R, the excess represents the amount of unsatisfied demand. The expected amount of unsatisfied demand is:

where g(x) represents the distribution of Lead Time Demand.

R

dxxgRx


A Retail Stocking Problem Daily demand (7 days/wk) is normally

distributed with mean = 60, standard deviation = 30.

Orders can be placed at any time, and will be filled in exactly 6 days.

It costs $10 to place and order, and each unit costs $5.

Annual holding costs are 10% of the value of the item.

We want to determine an efficient inventory policy that allows us to satisfy 99% of demand from inventory.


936

50$.

10$3656022

h

DSQ

48.73630 LL

First, we need to determine how much to order at a time:

Thus, our average cycle stock is 936/2 =468. In order to determine the re-order point, we

need to know the length of the lead time (6 days), and the standard deviation of lead time demand.


127.0

48.73

01.9361

L

PQzE

Now, to satisfy 99% of demand from inventory, we need:

From the table in the lecture note, we can see that E(z) = .127 implies that z = .77 (or so).

We can now calculate the re-order point, which consistsof expected lead time demand plus safety stock.

417)48.73(77.660 LzdLR

Batch Production


Suppose a machine produces a product at a production rate = p; e.g., p = 200 units/day.

Suppose the demand rate of this product is d; e.g., d = 80 units/day.

Since p > d, inventory will increase at (p-d) or (200-80 = 120) units/day.

Suppose the current inventory is 0. In 10 days, the inventory level would be 10

days * 120 unit/day = 1,200 units. In 20 days, the inventory level would be 20

days * 60 unit/day = 2,400 units. etc.


Suppose the machine produces a batch of this product, then stop, then resumes production at some later time when the inventory of this item is low. This is call batch production.

Batch production is very common in industry. When a machine is used to produce two or more

products, one product at a time. One decision the production manager has to make is

when to start producing each product, and when to stop.

The run time is the amount of time the machine is producing a batch. For example, producing at 200 units/day, if we want to

produce 2,000 units per batch, the run time is 2,000/200 = 10 days. Here, batch size Q = 2,000 units, the run time t = 10 days.


Maximum Inventory Level If the current inventory level is 0, what is the

inventory at the end of the run time? Since inventory will be rising at (200- 80 =120)

units/day, in 10 days, the inventory level will be 10 days * 120 unit/day = 1,200 units.

The inventory at the end of the run time is the maximum inventory. It is equal to (p-d)*t = (200 units/day - 80 units/day)*10 days = 1,200 units.

Why is it that the machine produced 2,000 units in 10 days, and the maximum inventory level is only 1,200? Answer: We consumed d*t = 80 units/day * 10 days =

800 units. After completing a batch, how long will it take to

deplete the inventory? Answer: It will take (p-d)*t/d = 1,200/80 = 15 days to

deplete the inventory. This is the off-time.


Number of Runs Per Year If the annual demand of this product is

D = 24,000 units, how many runs of this item do we produce each year? Answer: Since we are producing Q = 2,000

units per batch, there will be D/Q = 24,000/2,000 = 12 batches per year.

In other words, there are D/Q = 12 cycles per year. In each cycle, there is a period of time the machine is producing the product (the run time), and a period to allow the inventory to deplete (the off time).


Average Inventory What is the average inventory level?

During the run time, the inventory level rises from 0 to the maximum level of (p-d)*t = 1,200 units (see page 4).

During the off time, the inventory level drops from a maximum of (p-d)*t units to 0.

The average inventory level therefore = [0 + (p-d)*t ]/2 = (0 + 1,200)/2 = 600 units.

Since p*t = Q, then t = Q/p. We can rewrite the expression for the average inventory as (p-d)*t/2 = (p-d)*(Q/p)/2 = (1-d/p)Q/2.


Tradeoff Batch size = production rate * run time. Large batch

size means long run time, and high average inventory.

On the contrary, if the batch size is small, the run time is short, and we need to run many batches per year.

What is the average inventory if the batch size equals the annual demand D = 24,000 units? How many batches do we have to run per year? Answer: The average inventory = (1-d/p)Q/2 = (1-

80/200) (24,000)/2 = 7,200 units. We need to run one batch per year.

What is the average inventory if the batch size equals the weekly demand of 480 units (assuming 50 weeks/year)? How many batches do we have to run per year? Answer: 144 units; Run 50 batches per year.


pdH

SDQ

1

2*

Optimal Tradeoff Suppose the cost to carry one unit of inventory for

one year is H. Since the average inventory level is (1-d/p)Q/2, the annual inventory-carrying cost is H*(1-d/p) Q/2.

Suppose the cost to set-up the machine to produce a batch is S. Since we need to run D/Q batches per year (see page 5), the annual set-up cost is S*D/Q.

Adding the two costs, we have H*(1-d/p) Q/2 + S*D/Q. Using calculus, the optimal batch size is


Illustration Annual demand D = 24,000 units. Production rate p = 200 units/day. Demand rate d = 80 units/day (25

days/month). Set-up cost S = $100. Inventory-carrying cost H =

$2/unit/year.

The optimal batch size =2,000 units.


The machine will produce D/Q = 12 batches a year.

Run time t = Q/p = 2,000/200 = 10 days. Maximum inventory = (p-d)*t = (100-40)*20

= 1,200 units. Off time = 1,200 units/ 80 units/day = 15

days. In other words, the machine will run this

product for 10 days, stop (to do something else) for 15 days, before running this item again.

The New Boy Problem

Johnson & Pike, 1999


A-B-C Classification

% of total number of SKUs

% of total annual$ usage

20 40 60

80100

80

A specific unit of stock to be controlled is called a Stock Keeping Unit (SKU)


20 % of SKUs account for 80 % of total annual usage

Large number of cheap items A few very expensive items

1. A (most important) [ First 5-10 % of items]

2. B (intermediate important) [ 50 % of items]

3. C (least important) [40-45 % of items only ~ 20 % of value]

Inventory Management: Then and Now


Innovations in information technology and computer networking

tracking customer demand production ~ demand

1. Electronic Data Interchange2. Efficient Consumer Response (ECR)3. Vendor Managed Inventory (VMI)

How to utilize available information ?


Electronic Data Interchange Computer to computer transmission of

data (orders, invoices, payments, etc.) Fast and reliable tracking of inventory

levels, outstanding customer orders, backorders.

Shorter lead times for order processing, more reliable due-date quotation.


Efficient Consumer Response (ECR) Distributors and suppliers work

together so that information and goods can be exchanged quickly, efficiently and reliably Efficient store assortment Efficient replenishment Efficient promotion Efficient product introduction

Wegmans, Spartan Stores, HP, IBM, Compaq


Vendor Managed Inventory (VMI)

Supplier manages the inventory on it’s customer’s shelf (when and how much to order)


Framework for Inventory Management

Large number of items Large manufacturer ~ 500,000 items Retailer ~ 100,000

Items show different characteristics Demand can occur in many ways:

Unit by unit, in cases, by the dozen, etc.


Decision making in production and inventory management involves dealing with large number of items, with very diverse characteristics and with external factors.

We want to resolve: How often the inventory status (of an item)

should be determined ? When a replenishment order should be

placed ? How large the replenishment order should

be ?

Documents

Inventory Management Henry C. Co Technology and Operations Management, California Polytechnic and State University