Inventory Control by Woolsey and Maurer

8/20/2019 Inventory Control by Woolsey and Maurer

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INVENTORY CONTROL(FOR PEOPLE WHO REALLY HAVE TO DO IT)

Robert E. D. Woolsey, Ph.D., F.I.D.S.

and

Ruth Maurer, Ph.D.


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©2000 Lionheart Publishing, Inc.

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Preface: Questions and Answers About This Book 2Chapter 1. ARequiem for the EOQ 4• The Economic Order Quantity Model• Learning the Model and the First Experience of Reality• Killing the EOQ Vampire, the Silver Stake Method• The Cost of the Item• The Holding Rate

• The Setup or Ordering Cost• The Annual Demand or Annual Requirement• ReferencesChapter 2. Lot Sizing Methods of Inventory Control 14• What This Chapter Is For• The Economic Order Quantity Method• Periodic Order Quantity• Part-Period Balancing• Dynamic Programming• The Method of Silver and Meal• A Better Silver and Meal Method• Silver and Meal Quick and Dirty Fill-In-The Blank Inventory Form• First Silver and Meal Nomogram Example

• Second Silver and Meal Nomogram Example• Silver and Meal Nomograph Example• Greening’s Nomograph for Forecasted Demands• When Do I Use Which Method?• ReferencesChapter 3. The Woolsey Never-Fail Spare Parts Reduction Method 37• If You’re Using the EOQ, At Least Do it Right!• Setting the Scene• With the Computer Jocks and What Happened There• The Long Awaited Recalculation and What Happened Then• Icing on the Cake — Recalculating the Order Point• Final Warnings and Suggestions• Flowchart of the Woolsey Never-Fail Spare Parts Reduction Method• Inventory Example Problems• The Learning OrganizationChapter 4. El Pistolero del Inventorio 45• La Problema y Zopilote• Venustusiano Oso• Una Pregunta por Las Gerentes• El Neuvo Metodo• El Rey• Mas Preguntas de Importancia• Reference

1 • Inventory Control (For People Who Really Have to Do It) • eBooks Series

Contents

Inventory Control(For People Who Really Have to Do It)

Volume II in the Useful Management Series

By Robert E. D. Woolsey, Ph.D., F.I.D.S.And

Ruth Maurer, Ph.D.


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Preface

Questions and Answers About this Book

What is this book for?

This is a cookbook of quick and dirty methods for solving problems in real-

world, no B.S. inventory control.

Who is this book for?

This book is for bottom- to middle-level managers, third-world countr ies, and

small business where computers are either too expensive or labor is too cheap or too

uneducated to justify anything but the use of common sense.

It is also aimed at unpretentious community colleges and business schools that

want to teach people something they can use!

How smart does the reader have to be?

Smart enough to know that there are no pure technical problems in this world.

There are just technical problems imbedded in political problems.This book is ded-

icated to the idea that if you don’t deal with the political problems as well as you

deal with the technical problems, the polit ical problems will deal wi th YOU!

Does this imply that politics will be covered too?

Big time! The first author starts the book by taking on the most used (wrongly)

method in the inventory world called the economic order quanti ty. He points out

that this method (like any other) depends on data you can trust, such as deliverydates and the real need dates seldom given you by the sales types. He also tries to

warn the unwary reader that every method carries a bag of assumptions that need

to be carefully addressed before embarking on any method. A number of chapters

on political aspects of inventory control based on the experiences of the authors are

gleefully included. These chapters are more important than the others.

Do I need calculus, algebra, or statistics for this book?

No, common sense only is required.

Does this book come with a computer program or a diskette?

NO. This book is almost totally made up of certi fiably obsolete inventory meth-ods that no up-to-date large corporation or agency would even consider using.

They certainly can be programmed but the approach of the book is for the by-hand

user to have something theycan use and understand.

I’m an assistant professor at a business school, can I use this book?

YES, if you ever met a payroll in the real world.

Otherwise, NO!


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Thi s book is only an outli ne. Actual experience is an absolute requirement to f lesh out this book enough for classroom use . Also if you want to get promoted, and are

resident at a school with pretensions to academic respectability rather than utility,

you don’t want to be found dead in a field with this book!

Have the authors done this stuff for money in the real world?

You bet! The first author has only worked on a money-back guarantee plus inter-

est on payments at the prime rate if not satisfied. No requests for refunds to date.

He has worked for over 40 of the Fortune 400 and has now worked in America,

Australia, Canada, Denmark, England, Israel, Macao, The Netherlands, New

Zealand, South Africa and Sweden.

Has the first author ever really blown it?

Big time! In those cases no fee or expenses were required or accepted.Anyone who

says they never make a mistake in this field lies about hi s sex life TOO!

Why did the authors write this book?

Let’s face it, the right answer for inventory control is Just-In-Time and KANBAN

approaches. Unfortunately, attempts to move these methods in toto from Japan

have been less than successful. These methods, in our opinion, simply require (for

perfection) that you own, or control by any means possible,your suppliers (who are

also physically close) and that you have a well-educated homogeneous work force.

In third-world countries such as most of the United States, watching the mixture of northeastern city people and southern crackers (such as the first author) try to be

homogeneous in the typical l ight-industry workplace is futile. Thus, we propose

methods that even civil servants (such as ourselves) can use easily.

What are the authors really like?

Both authors have met payrolls,are opinionated, conceited, childish, intolerant of

pomposity (except in themselves), highly experienced, and by most crass, material-

istic measures: successful.

Preface


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The Economic Order Quantity ModelIn operations research, the most fundamental inventory model usually taught is

called the (Wilson) Economic Order Quantity (EOQ) Model. The purpose of this

chapter is to demonstrate that this model’s requirements are virtually never met in

practice. I believe its continued indiscriminate use to be dangerous to your corpo-

rate health. In an art icle in the Production & Inventory Management Journal by

Osteryoung, McCarty and Reinhart [1], we find the following statement:

“The conclusions of this article lead to the usual empirical paradox. That is, the

EOQ model which is advocated in financial textbooks is being widely used as a deci-

sion-making tool in practice. Unfortunately, the assumptions necessary to justify use

of the model are not met.”

I believe that the above statement is dead right . The usual nonsensical assump-tions are of constant demand, constant carrying costs, constant price and unlimit-

ed storage capacity. Let’s face it, the EOQ model is taught by academics because

a) they have no experience of reality, and b) (most importantly) i t is easy to teach.

It is then used by businessmen because a) it’s what the professor taught them, and

b) it is easy to use. This chapter is proposed as, hopefully, a silver stake through the

heart of the Economic Order Quantity Model. I will recount here my own experi-

ence with learning,applying and discarding the EOQ model. If I can do this right,

this essay wil l (hopefully) be the end of the matter.

Learning the Model and the First Experience of Reality

Like most other people, I learned the model in college and, being young and stu-pid, I believed that it worked. (After all, the professor was convinced.) My first expe-

rience with inventory reality was when I was assigned, fresh out of Air ROTC as a

young lieutenant, to the 70th Organizational Maintenance Squadron, Little Rock

AFB, Ark. I rushed down to talk to the supply sergeant, confident that any organi-

zation as big as the Air Force was using the latest thing. I asked him to tell me how

he set the order quantities of the spare parts for the dozens of B-47s that were our

responsibil ity. I, of course, had litt le doubt that he used the EOQ.

He told me that he watched carefully the demand patterns on major subassem-

blies, and ordered as best as he could subject to the TO (technical order) involved.

A thorough reading of the TOs revealed nothing, so I hung around a short while

and watched what he did. It didn’t take me long to realize that he was operating onessentially the old two-bin system,with emergency procurement parts handled sep-

arately. Clearly, it was my duty to reform the supply system of the Air Force, start-

ing with this sergeant.

Chapter 1(Political)

A Requiem for the EOQ


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The next day I showed up and with pencil and paper drew the usual graph of theOrder Quantity, Q, versus time, giving the well-known sawtooth wave shown in

Figure 1.

Figure 1

I explained that if we started with Q items in inventory and used them up at some

(known and deterministic) rate, we would run out at time T. However, at the lead

time, we would launch another order that, surprise, would arrive exactly at t ime T,

and so on and on. Clearly, if we start with Q items, go down to zero, restart with Q

items and go down to zero, and etc.,all we have to do is draw a line at Q/2 through

the diagram, as shown above, to get the average inventory. The next step was to get

him to agree that if we had a cost C of the part and a holding rate of I, then CIQ/2

was the average annual holding cost of the item. Clearly,as the number of items in

the order quantity Q increased, the cost increased. I then walked him through the

setup or ordering cost part. If it cost the Air Force S bucks to launch an order, andthe order size was the requirement R,divided by the order quantity Q, then clearly

S(R/Q) would get smaller as Q increased. I then wrote out, just as I had been taught,

the classic formula below.

He had no trouble with the idea that the minimum cost had to occur where the

rising holding cost line crossed the falling ordering cost line, and that it would be at

the lowest point of the total expected cost line made up of the sum of annual hold-

ing and ordering costs.

I was delighted; victory was in my grasp. Then things turned to #$%@. He askedme how I got the good old EOQ from the formula. I told him that all we had to do

was to take a derivative of the total expected cost formula, which I did on the spot.

He looked a bit uncertain, and politely asked (after all I was a second lieutenant)

how I knew that formula was right. I cheerfully answered that all we had to do was

to take a second derivati ve . He then, still politely (I was still a second lieutenant)

informed me that he didn’t know what in Hades a first derivative was, much less a

Q

Q/2

O T T T T T

Chapter 1

TEC = CIQ SR2 Q +


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second one. At this point I did what I should have done in the first place. I shut up,thanked him for his time,and learned to do it his way,before making any more hot-

dog suggestions for improvement.

Killing the EOQ Vampire, the Silver Stake Method

Many years have passed, but the experience is still fresh in my mind. Come with

me now as I take this model apart l ike a clock, and perhaps convince you never to

use it again. First, let us write the usual formula for the EOQ as follows, using an

appropriate reference that doesn’t believe it either: Woolsey [2]. We define as above

the TEC, total expected cost, as the sum of the annual holding cost plus the annual

ordering cost, or:

Where,as usual:

C is the cost of the item in $/unit,

I is the holding rate in % of price/i tem/unit time,

S is the set-up or ordering cost in $/order,

R is the annual requirements in units, and

Q is the order quantity in # items/order.

In the rest of thi s chapter, I am going to replace any symbol about which we have

some doubts wi th a LARGE question mark. Note that Q is a variable depending onall the others (we will only concern ourselves with the “known” constants). We will

now take each of the known constants, in turn, and treat them with the amusement

they deserve.

The Cost of the Item, C

Let us begin with C, the cost of the item, as our first doubtful symbol. For open-

ers, I always get amused when the costs are given in most textbooks without any dis-

cussion of first-in-first-out, last-in-first-out, lower-of-cost-or-market, or how-we-

do-it-here costing. It is always assumed that the cost accounting system is irrelevant

for lot-sizing purposes. Secondly, I am a firm believer that theonly real measure is

dollars after tax at present value,adjusted for inflation. For those of you with shortmemories, let me recall for you the time in the good old United States of America

when a peanut farmer was president and:

The inflation rate was 20+ percent.

The prime rate was 20+ percent.

The catalogs no longer came with price lists.

When you called for a price and they gave you one,

it was good only for a short time.

When you ordered and they billed you, it wasmore .

Chapter 1

TEC = CIQ SR2 Q +


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Now this little inflation factor becomes even more important among our readersin countries such as Bolivia, Israel, and Argentina, where people have been paid

twice a day so they could spend it before the exchange rate went down some more.

Trying to use an EOQ in such countries, even as close as Mexico,can be a terrifying

experience; I don’t recommend it. On the basis of this argument we award our first

question mark to C and our formula above becomes:

The Holding Rate, I

We now come to the well-beloved holding rate, usually given as a percentage of

the price per item per unit time. I have been greatly amused for over two decadesnow when the typical textbook graph of the total expected cost of the EOQ (plot-

ted against the order quantity) looks like Figure 2 below.

Figure 2

Chapter 1

TEC = ?IQ SR2 Q +

TEC

TEC*

TEC

Error in TEC*

Q Q 2Q*Q 1

SR/Q

CIQ

Error in Q*


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It is immediately seen that the TEC curve is very flat in the area of the optimalorder quanti ty,Q*, because of the small angle of the holding cost line usually shown

in most textbooks,as in the above picture. In this case,we can assume that we could

be in error around the optimum Q* between the points marked Q1 and Q2. Note

the wonder of the result; a large error in the constants that make up the optimal

order quantity Q*, when reflected on the TEC line, is seen to be of minimal effect

on optimum total cost TEC*.

In short, a big error in Q seems to result in a small error in TEC. This seems to

imply that if you feed this formula some garbage values of C,I,S,or R,it really won’t

hurt you very much in terms of total expected cost. There is only one other thing in

the world that you can feed such garbage to and get something good out of; it’s

called a pig. If you feed a pig garbage, eventually you get bacon. However, it isimportant to note that the creation of bacon requires total commitment from the

pig. Total commitment to the EOQ may be equally fatal to your profit s. We will show

this next.

I argue that the variance of the holding rate is more than most people imagine.

Most people, and businesses, tend to assume that the holding rate is the cost of

money, i.e., prime rate plus points. I believe this to be a bloody dangerous assump-

tion,and I will demonstrate it. I believe that there are really only four kinds of prod-

ucts that most businesses really care about. These products are measured only by

market demand and profit margin or markup as shown below in Table 1.

Table 1: The Four Types of Products

Type M kt. Demand M arkup Translation

1 High High Customers beating on doors, profit obscene

2 High Low Customers beating on doors, profit zip

3 Low High Nobody wants it, profit obscene

4 Low Low Nobody wants it, profit zip

One doesn’t have to be too smart to agree that, if you could, you would only stock

type 1,high market demand and high markup. Let us now realize what else this lit-

tle classification tells us. It really says that, given a choice, we would li ke to stock the

parts wi th the biggest markup in an expanding market.The next step, however, once you buy this argument, is a killer. It really implies

that the minimum holding rate is the rate of return you are getting on your highest

markup item in an expanding market, because, if you could, you would put every-

thing there (but you can’t) .This means that far from being the prime rate,the hold-

ing rate is lower bounded by your highest markup on your best sell ing product. In

short, the line you see for holding cost in most textbooks is usually drawn like

Figure 2, reflecting cost of money, or prime rate. Clearly, you should be using a

Chapter 1


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holding rate greater than your cost of money or you have no business being in business.

Say that we discover that our best mover in our product l ine, which also makes a

bundle after tax at present value, has a markup of 60 percent. Let’s now redraw our

graph above with the new holding rate but using the same ordering cost line as

before. This is shown in Figure 3.

Figure 3

From the above graph,we note that a) our minimum Total Expected Cost ismuch

higher, and b) with the same amount of error as in the previous graph,we can be in

deep trouble with its effect on total cost . I hope that I have now convinced you that we

Chapter 1

TEC*

TECCIQ/2

SR/Q

Q Q 1 Q 2Q*

Error in O"

Errorin

O"


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now have so much variance in our holding rate, I, that we should replace it with theappropriate question mark in our formula above, result ing in:

The Set-Up or Ordering Cost, S

With this one, I usually get strong arguments like:“Why,we havethat data to four

decimal places; it comes from job standards taken by our industrial engineers.” Now

let us say that we have a happy inventory clerk working efficiently in his tool crib.

He is approached by an industrial engineer who says, somewhat speedily:

“Hitherehappyinventoryperson,I’myourlocalindustrialengineerheretodoatimeand

motionstudytoincreaseyoursafetyandproductivity, just ignore me .”While the clerk is trying to figure out what he said, the IE produces his clipboard

and punches his stopwatch to time the next order that has just come in. Now I am

here to tell you as a senior member of the Insti tute of Industr ial Engineers that I

know what will happen next. That clerk wil l (particularly if unionized), before your

very eyes, turn into a good imitation of Mikhail Baryshnikov doing Swan Lake, and

make every movement at this speed or slower. This, ladies and gentle-

men, is what you have to four decimal places as job standards.What you really have

to four decimal places is nonsense. The only way to set real job standards is to do i t

yourself, or take your data from behind a distant mountain wi th binoculars. On this

basis, I award our third question mark to our ordering cost, S, resulting in:

The Annual Demand or Annual Requirement, R

Showing that the annual requirement is deserving of a question mark is really

shooting fish in a barrel. Let’s face it, weknow where the requirement comes from.

It is extracted by the right- or left-hand rule from some appropriate orifice by the

marketing department. Everybody knows that marketing routinely inflates their

demand forecasts. Let us, for the inexperienced, discuss why this happens. It should

be clear to anyone with production experience that where you stand depends on

where you sit . A sales manager, when asked by his CEO about his projection for the

next quarter’s sales,must say something like this (or lose his job):“Sales, sir, will be UP, and I mean UP, we’re going to blow the competition

AWAY!”

Now let’s imagine for a moment, that the sales manager has an honesty attack and

tells the ultimate biggy something like:

“Well, boss, we are going to be hanging on by our fingernails this quarter, our

competition is going to eat us ali ve .”

Chapter 1

TEC = ??Q ?R2 Q +

TEC = ??Q SR2 Q +


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This statement would be followed by a gunshot, the sound of a Winchester beingrecocked and the CEO saying ominously, “Bring me another sales manager.”

Now there may be a few marketing types out there who always tell the truth to

production, and I humbly apologize to those two. Any old production person

knows that he is constantly ground between the upper and nether millstone of

a) sales and b) accounting. For example, occasionally the sales manager will appear

with the ult imate biggy, look at the production floor and loudly proclaim to all and

sundry:

“Why we can’t sell from an empty wagon, we’ve got to fill our warehouse UP!WE

NEED MORE PRODUCTION!”

Under the beady eye of the CEO, you respond by making more of what the jerk

cannot possibly sell,with the usual result of visible in-process inventory on the floorand in the warehouse. You are hardly surprised that some months later, this stuff

(also still on your profit and loss sheet) is still there. The sales manager that caused

it all is, also a surprise, incommunicado.It is not long after you do this that you get

a visit from the comptroller. It is important to remember that accountants only get

promoted when they kill things. Through his green eyeshade he views this invento-

ry and, in a puff of smoke, turns into Clint Eastwood hissing the following words:

“Well now, we need to first write off this inventory and then we will see about

writing off the production manager that was stupid enough to make it. That’ll real-

ly make my day .”

Have you ever noticed that you never see sales managers and comptrollers at the

same time?They are like the little plastic weather houses I grew up with. When theweather was good, the children came out, and when it was stormy, the witch came

out.But it was impossible for them both to be out simultaneously.

What all of the above really means is that the annual requirement will be all over

the map, due to the above political problems and Woolsey’s law of the forecast [ 3],

which is:

1.The forecast iswrong , and

2. It will change !

On the basis of the above arguments, I award our last question mark to the annu-

al requirement R, result ing in:

At this point in time, we look at the above formula and see if there isanything we

can rely on.Yes, there isone thing left that we can believe in. It obviously isn’t the Q

because after all Q is, for cat’s sake, the VARIABLE. This means that in that whole

formula above, there is onlyone constant, one truly solid assumption,one thing that

Chapter 1

TEC = ??Q ??

2 Q +


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is unchanging through the ages, yea verily, like unto the Rock of Gibraltar, it is (areyou ready for this), the:

2

But wait, that 2 is there because we have made the heroic assumption that we have

constant demand , which means that the demand pattern looks like the picture we

started this whole thing with (see Figure 4 below).

Figure 4

Has anyone who is reading this in their l ifeever seen a demand pattern that looks

like that ?I have now worked and taught on five continents,and I have yet to see an

example of constant demand. We know perfectly well that demand patterns looks

more like a graph of earthquakes measured on the Richter scale than the graph

above.After all, how often have you seen your marketing types penalized for giving

you nonsense production forecasts?We experienced types also know that the zero

on the graph above is really somewhere below the bottom of this page, the differ-ence being known as, dare I say it, SAFETY STOCK. Under these circumstances, if

you want to get Q/2,all you have to do is close your eyes and draw a line anywhere

on the graph. It really won’t make any difference, trust me.

So it turns out we can’t even trust the 2 . At this point we award another question

mark to the 2,and a final one to the variable Q, giving the desired result:

Recommendation

If you, or your firm are using the EOQ model, does it occur to you that an

accounting audit of costs might be in order?If you continue to love and use the EOQ wi thout knowing what it i s costi ng you, I can only suggest that you deserve each other.

Chapter 1

Q

Q/2

O T T T T T

TEC = ??? ??? ?+?=


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References

1. Osteryoung, J. S., Nosari, Eldon, and McCarty, Daniel E., “Use of the EOQ

Model for Inventory Analysis,” Producti on and Inventory Management, Vol. 27,

No. 3, (1986), pp. 39-46.

2. Woolsey, R.E.D., and Swanson, H.S., Operations Research For Immediate

Application, A Quick & Dir ty Manual, New York, Harper & Row Publishers,

(1975), pp.39-41.

3. Woolsey, R.E.D., and Lienert, C. E., “Ordering Inventory When The Forecast Is

Ridiculous,” Producti on and Inventory Management , Vol. 27, No. 1, (1986), pp.

144.

The above paper appeared originally as:4. Woolsey, R.E.D., “A Requiem For The EOQ: An Editorial,” Production and

Inventory Management ,Vol. 29, No. 3, Third Qtr,.1988, pp. 68-72.


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What This Chapter is ForThis chapter will display five different methods for lot sizing inventory to mini-

mize total inventory costs. These methods are :

Economic Order Quantity

Periodic Order Quantity

Part-Period Balancing

Dynamic Programming

Silver and Meal

Each of these methods has a number of assumptions associated with their use

that are seldom, if ever, met in practice. In this time of material requirements plan-ning,MRP II, OPT, Just-in-Time and KANBAN, why are such obsolete, highly lim-

ited methods being presented here?Because the author believes that often their

application is better than the present system that they might replace. In addition,

they attack slightly different problems than do the newer methods.

Readers are hereby cautioned that application of any of the above methods

requires first a reading of the previous chapter on the political aspects of lot sizing,

especially with the EOQ method. In that chapter, the reader will have discovered

that a high order of cynicism is an absolute requirement when confronted with the

usual data requirements of any of the methods.

The first four methods above are presented in order of increasing complexity and

increasing accuracy. EOQ is the easiest to teach, to understand and to use incor-rectly. It also has the greatest potential to bedead wrong . The next-to-last method,

dynamic programming,is mathematically and computationally complex,expensive

to use, but is absolutely optimum, if you agree that the forecast will NOT change ! My

experience tells me that the two laws of the forecast are:

The forecast is wrong!, and

The forecast will CHANGE!

To expect the forecast not to change in the workplace is roughly equivalent to stat-

ing that, using calculus and neglecting air resistance, the cannon ball will fall right

there . Any old artilleryman will tell you the safest place to stand is where calculus

tells you the ball will fall. This is because by assuming away air resistance, we have

also blithely assumed away reality.

The last method, that of Silver and Meal, is clearly the author ’s favorite. It assumes

that the forecast is wrong and will change. However, it is so simple to use that we

really don’t have to care. Silver and Meal do a rigorous mathematical derivation in

[1]. I will derive it using some mathematically shaky but reali ty-based common-

sense arguments based on many years of bitter experience.

Chapter 2

Lot Sizing Methods of Inventory Control


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Finally, if you can afford it, and you can control your suppliers and your sales fore-cast sufficiently, you shouldn’t touch any of the above methods with a barge pole

but should use a lot size of 1.

The Economic Order Quantity Method

This method is rightly considered the fundamental model in inventory control

and can, correctly used, sti ll generate more consistent results than rule of thumb

methods. The usual reference is to Harris [1]. This method makes the assumption

that we start off with Q items in inventory,over a given time,we will use these items

at a known and constant rate until we run out. Sometime before running out, we

note the bloodshot eyeball of the inventory control clerk looking ahead and realize

that we are going to run out! At some time before running out, the clerk launchesanother order. This tr igger point is a set level of inventory that will just last until we

run out, using the present demand rate. In short, the Trigger Point may be defined

as (Annual Demand)*(Lead time in days)/(365 days). In theory, if the clerk launch-

es the order on or before the lead time for it to arrive, this person’s backside is suf-

ficiently covered. It is, however, good to remember that the lead time for the order

to arrive is supplied by the vendor , who has every reason to l ie to you. If he or she is

sufficiently truthful about the extended time it will take to deliver your order, you

just might seek out another supplier . If the order arrives exactly at the time you run

out, you have the beloved sawtooth shaped curve seen below.

Figure 5

All that is required in the above situation is to note that if we start with Q in

inventory and go down to zero repeatedly,we can draw the average inventory line

seen above labeled Q/2.We now define that the cost of the item is C dollars per unit. We further define

that the holding cost per i tem per day (expressed as a percentage of the cost, C) is I.

For example if we have an item that costs $25 dollars with an annual holding rate

of 12 percent, and our order quantity is 40 items per order, the average holding cost

per year is:

Holding Cost = C * I * (Q/2) = $25 * .12 * (40/2) = $60

Chapter 2

Q

Q/2

O T T T T T


18/52


In short,we can express the annual holding cost as AHC = CIQ/2. It is instructiveto note the dimensions of the above holding cost explained below.

Holding Cost ($/year) = (Price ($/unit))*(Holding Rate ($/$ .year))*

(Q units/order)/(2 (1/order)).

As it is assumed that C and I are known, we just have the relationship Cost =

Constant * Q. This may be easily graphed with Q versus cost as shown in Figure 6.

Figure 6

What the above graph tells us is: “The more we hold in inventory, the more it’s

going to cost us in holding cost.” Now let’s derive the second part of the Economic

Order Quanti ty Model, the annual ordering cost.

Let us define that the cost to launch an order, get it in, and shelve it is S dollars per

order. Let us also define the forecasted demand for the year as D i tems per year. Now

if we order Q items in an order, it should be apparent that the number of orders per

year is:

# orders per year = (D items per year)/(Q items in an order).

And it follows,using the above example, that if we have an annual demand of 120items and an ordering cost of $20 per order, the cost of such annual ordering should

be:

Annual Ordering Cost = S * D/Q = $20 * 120/40 = $60.

Chapter 2

Cost

O Q


19/52


As before, we note the dimensions of the process as:Annual Ordering Cost ($/year) = (S ($/order))*(D units/year)/Q (units/order).

In short,we can express the annual ordering cost as AOC = SD/Q.As it is assumed

that S and D are known, we just have the relationship Cost = Constant/Q.This may

be easily graphed with Q versus Cost as shown in Figure 7.

Figure 7

What the graph in Figure 7 tells us is: “The more often we order, the more it’sgoing to cost us in ordering cost.” Combining the two graphs in one where Q is now

plotted against both costs, we create the combined graph in Figure 8.

Figure 8

Chapter 2

Cost

O Q

Cost

O Q


20/52

Q*=2*S*D

CI


Picking any points on the Q line, say Q1,Q2 and Q3,we add up the contributionof both the holding cost and the ordering cost to generate the graph for the Total

Annual Cost as shown in Figure 9.

Figure 9

The graph in Figure 9 tells us that we need to balance the contributions of hold-

ing and ordering costs to minimize our total cost. The minimum will obviously

occur where the increasing holding cost line crosses the decreasing ordering cost

line. Put another way, this is where CIQ/2 = SD/Q.However, solving the above expression for Q gives us the famous Economic Order

Quantity, Q*, or:

Let’s now take the following example, and apply the method above:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 10 10 10 10 10 10 10 10 10 10

We will use the values given above, which are:C= cost of item = $25/item

I= holding rate = 12 percent of cost/year

D= annual demand =120 items

S= cost of order = $20/ order

Using the formula: TEC = CIQ/2 + SD/Q, we get:

TEC = (25 * (.12)/2)*Q + 20 * (120/Q) = 1.5*Q + 2400/Q

Chapter 2

Cost

O Q


21/52

Q*=2*20*120

25*.12= 40


Using the formula above for the EOQ we get:

This tells us that the cost minimizing lot size is 40 items an order. Now let’s check

this month by month to confirm that this is the right answer:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 10 10 10 10 10 10 10 10 10 10

Order 40 0 0 0 40 0 0 0 40 0 0 0

Start 40 30 20 10 40 30 20 10 40 30 20 10

End 30 20 10 0 30 20 10 0 30 20 10 0

Average 35 25 15 5 35 25 15 5 35 25 15 5

We see at once that as we order three times, our annual ordering cost is 3

orders/year * $20/order = $60/year. Our annual average holding cost is

(35+25+15+5+35+25+15+5+35+25+15+5)/12 = 20 items/year * $25/item * .12 =

$60/year.

Political Discussion

The fundamental problem with this method is the assumption that we have con-

stant demand. In other words,we assume the sawtooth illustration we saw before is

correct. The hard facts are,however, that you will be hard pressed to find a forecast

that is not subject to, at least seasonal trends, such as the example below:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Using a cost of $2 per unit per month,a ordering cost of $300 and an annual total

demand of 1105, the optimum EOQ is found to be Q* = 166.Applying this to the

above problem as before we get:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 166 166 223 270 230 166Start 166 156 146 131 111 207 250 270 230 166 126 126

End 156 146 131 111 41 27 0 0 0 126 126 116

Average 161 151 138.5 121 76 117 125 135 115 146 126 121

Total Cost = Ordering Cost + Holding Cost = 1,800 + 3,065= $4,865

Chapter 2


22/52


We note that we order six times at $300/order, so our ordering cost is $1,800.Butour average inventory is now 1,532.5, which gives an annual average holding cost of

$2*1,532.5 = $3,065. We notice at once that our happy assumption that the mini-

mum cost will occur when the increasing holding cost line crosses the decreasing

ordering cost line is violated.This is becausewe no longer have satisfi ed the necessary

assumpt ion of constant demand . In short the“ lumpiness” of demand is doing us in.

We need some way to smooth out this “ lumpiness.” Thus, the next method.


We can often reduce inventory carrying costs by finding an economic time inter-

val between orders. Divide the EOQ found above which is ______, by the mean

demand rate (total demand divided by the number of periods),which is in this case1105/12 = _____.Doing this, we get _______ months.We can just round this up to

___ months and apply it to the example below:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 20 35 250 520 270 10

Start 20 10 35 20 250 180 520 270 270 40 10 10

End 10 0 20 0 180 0 270 0 40 0 10 0

Average 15 5 27.5 10 215 90 395 135 155 20 10 5

Total Cost = Ordering Cost + Holding Cost = 1,800 + 2165 = $3,965

The first thing we notice is that we are still ordering six times, but our average

inventory has taken a drop to 2,165 from 3,065. In short, by doing one divide, we

have realized a reduction in cost of almost 20 percent (actually 18.5 percent).

However, we may do better with a little more computation overhead using the

next method.

Part-Period Balancing

This method simply says: “Keep increasing the number of periods you are order-

ing for until your holding cost is as close as you can get to the ordering cost.” Let’s

see the example above again:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Let’s recall that our ordering cost is $300/order. If we were to order only for the

first month, our holding cost would be $2/item/month *10/2 average items = $10.

As this is nowhere close to $300,we consider ordering for two months. If we ordered

the second month’s demand of 10 to come in at the first of January, we would hold

Chapter 2


23/52


it all of January and half of February, which gives a cost of $2/i tem/month * 3/2 *10 = $30. But to get the total cost for ordering for two months, we would have to

add in the $10 we got before for January’s demand,giving a total of $10 + $30 = $40.

So for up to now we have:

Ordering for one month is $2 * 10 * 1/2 = $10.

Ordering for two months is $10 + $2 * 10 * 3/2 = $40.

Clearly, ordering for three months would give:

$40 + $2 * 15 * 5/2 = $115.

And ordering for four months would give:

$115 + $2 * 20 * 7/2 = $255.

And ordering for five months would give:

$255 + $2 * 70 * 9/2 = $885.Now as ordering for four months is $255, which is closer to the ordering cost of

$300 than ordering for five months for $885, our first order would be for four

months and 55 items.

Using the space below, calculate the other orders and enter them in the table

below and calculate the expected holding and ordering cost for this method.

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 55 0 0 0 _________________________________

Star t 55 45 35 20 _________________________________

End 45 35 20 0 _________________________________Average 50 40 27.5 10 _________________________________

If you did it right you get the display below:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 55 70 180 250 270 270 10

Start 55 45 35 20 70 180 250 270 270 40 0 10

End 45 35 20 0 0 0 0 0 40 0 0 0

Average 50 40 27.5 10 35 90 125 135 155 20 0 5

Total Cost = Ordering Cost + Holding Cost = 2,100 + 1,385= $3,485

In this case,we note that we now have ordered seven times for an ordering cost of

$2,100, and we have incurred a holding cost of $1,385 for an additional reduction

of 12.1 percent.Notice that the computational burden has increased substantially to

obtain this difference. Economists have a name for this, it’s called diminishing

returns . Let us now discuss the method that will get us the optimum solut ion, usu-

ally known as the Wagner-Whitin method [2], or dynamic programming.

Chapter 2


24/52


Dynamic ProgrammingWhat this method does is to implicitly examine every possible way to order the

inventory and then choose the best of these. This method requires that you fill in

the table shown below. Note that for a 12-period forecast for one item you might

have, in a worst-case scenario, to fill in 144 entries in the table below. This means

that if you were using this method to lot-size 36,000 items for a year that you might

have to make calculations to fi ll in 36,000 * 144 or 5,184,000 entries. Also, the real-

ly bad news is that you have to do the whole thing over again every time ANY sin-

gle period’s forecast changes . The good news is that the method gives the optimum

solution, the bad news is that the computational burden is ferocious. I will cheer-

fully admit that there have been many advances in reducing this computational bur-

den. For examples, see theProducti on & Inventory Management Journal issues for1990-1993. However, the hard facts, in my opinion, are that this method is out-

standing in theory but hard to understand, and virtually impractical in the real

world.

Let’s start filling in the table as follows.

The intersection of column one and row one is the inventory cost to order for one

month in period one.We should recall from part-period balancing that this is $300

to order plus an average holding cost of $2 * 10 * 1/2 for a total of $310.

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

1 310 620 655 735 925 1405 1955 2525 3055 3395 3175 34852 340 665 715 945 3175 3395 3505

3 415 765 1065 3175 3445

4 555 1255 3245

5 1110 x

6 x x

7 x x

8 x x

9 x x

10 x x

11 x x

12 x xOrder 55 0 0 0 70 180 250 270 280 0 0 0

The intersection of column one and row two is the inventory cost to order for two

months in period one. As this is just $310 plus the addit ional holding cost of $2 *

10 units held for 3/2 of a period we get a total of $340.

The intersection of column one and row three just adds to the above $340, the

additional holding cost of $2 * 15 units held for 5/2 of a period, a total of $340 +

$75 or $415.

Chapter 2


25/52


The intersection of column one and row four just adds to the above $415 theadditional holding cost of $2 * 20 units for 7/2 of a period, a total of $415 + $140

or $555.

The intersection of column one and row five adds in the non-trivial additional

cost of holding 70 items for 9/2 periods or $2 * 70 * 9/2,giving $1,110.Notice , this

incremental cost of $640 is greater than the additional ordering cost of $300 for

another order, so wecould stop further calculations down this column as the situa-

tion will just get worse.

The intersection of column two and row one should contain the cost to order for

one month in period two, assuming that you ordered for one month in period one.

This is simply $300plus to order, plus average holding cost of $2 * 10 * 1/2 or $310,

plus the cost of ordering 10 units in period one which is also $310 for a total of $620.We may then proceed as before until we step over our ordering cost of $300

as before. The intersection of column three and row one should be the minimum

cost to order for one month in period 3 given that you made the best decision for

ordering for the first two months. As our choices are to order for month one in

month one and month two in month two at $620 or to order for month one and

two in month one at $340, the choice is obvious. So we add to this $340 the cost of

ordering in month three (which is $300) and the holding cost (which is $2 * 15*1/2

=15) which gives $655. The reader is invited to check the other entries in the table

if he or she is so inclined.

Month 1 2 3 4 5 6 7 8 9 10 11 12Demand 10 10 15 20 70 180 250 270 230 40 0 10

1 310 620 655 735 925 1405 1955 2525 3055 3395 3175 3485

2 340 665 715 945 3175 3395 3505

3 415 765 1065 3175 3445

4 555 1255 3245

5 1110 x

6 x x

7 x x

8 x x

9 x x

10 x x11 x x

12 x x

Order 55 0 0 0 70 180 250 270 280 0 0 0

After the table is completely filled in, we start at the cost in the upper right hand

corner of the table (the entry in row 1, and column 12). We now look diagonally

down the table moving always down and to the left, looking for the cheapest cost on

Chapter 2


26/52


this diagonal. Once we have found it (i t’s $3,245) we enter the total items (= 280)for this last order into the order row at the bottom.We then look at row one in the

last column before the column that generated the previous order, and do the pro-

cedure again, fi lling in the orders as shown above.

Putting the above results into our usual table gives:

Month 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 55 70 180 250 270 280 10

Start 55 45 35 20 70 180 250 270 280 50 10 10

End 45 35 20 0 0 0 0 0 50 10 10 0Average 50 40 27.5 10 35 90 125 135 165 30 10 5

Total Cost = Ordering Cost + Holding Cost = 1,800 + 1445= $3,245

Political Discussion

Wehave found the optimal solution,but the computational cost is definitely non-

trivial. Further, as we must find out the optimal order now by going back to front,

this means that what we do now is a function of the forecast at the end of the fore-

cast period . It is important to realize that the further out we get in the forecast, the

greater the expected error. This method says go out to where the forecast is least reli-able, and figure out what to do now! I must tell you that I have real problems with

this concept. My experience tells me that the forecast is wrong and that i t will

change. If you agree with me, this says that the computational requirements to get

the optimum is going to be rarely, if ever, justified by the use of this method. But

there is good news yet, the method of Silver and Meal follows next.

Chapter 2


27/52


The Method of Silver and MealLet’s pause for a moment and let me tell you what I believe about inventory con-

trol. First off, lets see the graph of the EOQ again, shown in Figure 10.

Figure 10

Now, I do believe that the minimum cost for inventory situations occurs where

the ordering costs and the holding costs balance. I fur ther believe that as we get fur-

ther and further out in a forecast the expected error of the forecast is roughly equal

to the time period squared multiplied by the size of the forecasted order in that peri-

od. In equation form we could write:

Expected Error of the Forecast = T2*D(T)

I believe this because my experience tells me that errors in forecasts go up rough-

ly as the square of the time periods from now.And it is certainly true that the big-

ger the order in the future, the greater the probability that i t wi ll CHANGE!

Chapter 2

TEC*

CIQ/2

SR/Q

Q Q 1 Q 2Q*

Error in O"

Error in TEC"TEC*

TEC


28/52


Let’s take a look at the measure of the above equation; assume that the time peri-od is months and the demand is in tons/month. If this is the case the measure is:

Expected Error + Months2*Tons/Month = Months*Tons

It only remains to find some combination of the cost of the item (C), the holding

rate (I), and the ordering cost (S), that has the same measure of months*tons. We

don’t have to look far to find:

S/CI = Months-Tons

We therefore conclude that a rough and ready approximation to when we should

order is given by:

T2*D(T) >

In short, the above rule says:

Launch the order when the time period,squared, times the demand in that

period becomes greater than the ratio of the ordering cost to the holding cost.

Now your first reaction should be that no way in Hades could lot sizing be that sim-

ple and cover your backside.Let’s test i t on our well-beloved problem and see.Recall

that the first five months of forecasted demand look like:

Month 1 2 3 4 5

Demand 10 10 15 20 70

We first try T = 1,with a holding cost of $2/item/period and an ordering cost of

$300,which gives:

T2*D(T) > S/CI, or,

12*10 > 300/2

Clearly, the answer is no, so let’s try T=2, which gives:

22*10 > 300/2

Again the answer is no, so let’s try T=3, which gives:

32*15> 300/2

Once more, the answer is no, so try T=4, which gives:

42*10 > 300/2

The answer is yes! This tells us that we should order four periods demand or 55items to come in at the start of the first period. This is also the first answer found

by dynamic programming above.

The method above is really rough and ready, and should not be used for anything

other than a ball-park estimate of where the lot sizes are . Let’s see if we can’t define a

more representative model than the (admittedly) quick and dirty one above.

Chapter 2

SCI


29/52


ABetter Silver and Meal MethodIf we order for one month,and we assume that we count our inventory at the end

of the month, clearly our cost per unit time is:

TEC(T =1)= *(S )= *($300)=$300 /month

Because by the time we count the inventory at the end of January,we haveused

it, so we incur no holding cost at all .

If we order for two months on the same basis, our cost per unit time is:

TEC(T =2)= *(S+CI*D(T ))= *($300+2*10)=$160 /month

The first thing we notice here is that our cost per unit time has been almost cut

in half by ordering for two months rather than one.

If we order for three months,as before, our cost per uni t time is:

TEC(T= 3 )= (S+CI*(D(T) +2*(D(T +1))= *($300+$2*(10=2*15))=$126.66 /month

Again, we note that once more,our total cost per unit t ime drops.We should also

realize that i f we ordered March’s demand to come in at the first of January, that we

will count it twice , once at the end of January and once at the end of February.

Now ordering for four months, our cost per unit time is:

This time, we got only a very small reduction, but a reduction none the less.

Clearly,as long as our total cost decreases we should continue to march. So for five

months we know holding April’s 70 units at $2/item/month is going to increase our

costs,which it does, giving:

We quickly conclude that, to minimize cost per unit t ime,our first replenishment

should be for 55 units. (Sound familiar?)

From the above process, we may state a general formula which is:

TEC(T) = *(S+CI*(1*D (2)+2*D (3)+3*D (4)... +T*D(T+ 1 ))

Chapter 2

1T

11

1T

12

13

1T

TEC(T =4)= *(S+CI*D(T ))+2*(D(T+1)+3*(D(T+2))

= *($300+2*(10+2*15+3*20))=$125 /month

1T

14

TEC(T =5)= *(S+CI*(D(T ))+2*(D(T+1)+3*(D(T+2)+4*D(T+3))

= *($300+2*10+2*15+3*20+4*70))=$212 /month

1T

1

5

1T


30/52


Or, using summation notation, we have:

The stopping rule is simple.As soon as TEC(T+1) > TEC(T),STOP!However,we

don’t want to have to calculate however many steps it may take to find the optimum,

so let’s design a nomogram or a fil l- in-the-blank form to do it for us. To do this, we

must first modify the above general formula by dividing both sides through by CI;

this gives:

I hope that the reader will agree with me that dividing both sides of the above

equation by a constant shouldn’t change the stopping point. It will , however,change

the values of the total costs at various times by the factor of CI. Let’s test the above

problem again to convince ourselves that even though the costs change, we stop at

the same place.

If we order for one month, and we assume that we count our inventory at theend

of the month, clearly our cost per unit t ime is:

TEC(T =1)= *(S/CI )= *($300/2)=$150 /month

If we order for two months on the same basis, our cost per unit time is:

TEC(T =2)= *(S/CI+(D(T) )= *($300/2+10)=$80 /month

If we order for three months,as before, our cost per unit time is:

TEC(T =3)= *(S/CI+(D(T) +2*(D(T)+1))

= *($300/2+(10+2*15))=$63.33 /month

Now ordering for four months, our cost per unit time is:

TEC(T =4)= *(S/CI+(D(T) +2*(D(T)+1)+3*(D (T +2))

= *($300/2+(10+2*15+3*20))=$62.50 /month

TEC(T =5)= *(S/CI+(D(T) +2*(D(T)+1)+3*(D(T+2))+4*D(T+3))

= *($300/2+(10+2*15+3*20+4*70))=$62.50 /month

We therefore see that our stopping point is the same as before. To minimize cost

per unit time, our first replenishment should be for 55 units, just as before. Let’s

now use this modified method to design the promised fill-in-the-blank form.

Chapter 2

1T

11

1T

12

1T

13

1T

14

15

TEC(T) = * S + CI* Σ (j - 1)*D(j) 1T ( )( )

j=T

j=1

TEC(T) = * + Σ (j - 1)*D(j) 1T ( )( )

j=T

j=1

TEC(T) CI

SCI

1T


31/52

Silver and Meal Quick and Dirty Fill-in-the-Blank Inventory Form

This method assumes that you have a forecast for some number of periods in the

future for the item of interest. It further assumes that you have a decent idea of:

A) The set-up or ordering cost to place an order, and

B) The holding cost per unit per period to hold one unit of the item in invento-

ry for one period.

How to do it:

1. Divide the ordering cost by the holding cost and put it next to where it says

RATIO in column “0.”

2.Enter the forecasted demand in row B.3.Starting at the first column, multiply the entry in row A by the entry in row B

and enter the result in row C.

4.Add the entry in row C to the last entry in row D and enter the result in row D.

5.Divide row D by row E (the number of time periods) and enter in row F. (This

is simply the total average cost divided by the holding cost.

6.When the value in the F row increases, STOP! Sum up the demand to the peri-

od before the F row value increases and order that number of items.

A 0 1 2 3 4

B

C=A*BD=RATIO

E 1 2 3 4 5

F = D/E

Reference

Hesse,Rick and Woolsey,R.E.D., “Applied Management Science,” Science Research

Associates, Inc., Chicago; Palo Alto, Calif., 1980,pp.63-65.


Chapter 2


32/52


First Silver and Meal Nomogram Example

Now let’s test our Q&D form with the problem we have been using.

Period 1 2 3 4 5

Demand 10 10 15 20 70

A 0 1 2 3 4 5

B 10 10 15 20 70 180

C 0 10 30 60 280

D 150 160 190 250 530

E 1 2 3 4 5 6

F 150 80 63.33 62.5 106

It’s easy to see that the values generated coincide with those from the modified

formula above.

The next replenishment would look like:

A 0 1 2 3 4 5

B 70 180 250

C 0 180

D 150 330

E 1 2 3

F 150 155

This tells us we should order 70 as the second replenishment, if the forecast does-

n’t change! Continuing on with the form we would generate the orders as shown

below with associated costs.

Period 1 2 3 4 5 6 7 8 9 10 11 12

Demand 10 10 15 20 70 180 250 270 230 40 0 10

Order 55 70 180 250 270 280

Start 55 45 35 20 70 180 250 270 280 50 10 10

End 45 35 20 0 0 0 0 0 50 10 10 0

Averg. 50 40 27.5 10 35 90 125 135 165 30 10 5

Total Cost = Ordering Cost + Holding Cost = 1,800 + 1,445 = $3,245

We now see that Silver and Meal, in this part icular case , gets the optimal solut ion,

with a lot less pain and agony than dynamic programming.It can be argued that the

author has cleverly selected an example that supports his own point of view.Let us

have no doubts about this, the charge is true. I like this method because it allows you

to be able to react to changes in the forecast and i t tells you where the trigger points

are! Let’s consider the following example.

Chapter 2


33/52


Second Silver and Meal Nomogram ExampleSay we have the forecast below for the next three months, with a holding cost of

$3 per item per month and an ordering cost of $561.

Month 1 2 3

Demand 35 45 67

Plugging this into our Q&D we get:

A 0 1 2 3 4 5

B 35 45 67 77 120

C 0 45 134D 187 232 366

E 1 2 3

F 187 116 122

This tells us that our first replenishment is to order 80 units to come in at the start

of the first period.At this point a grizzled old foreman looks over your shoulder and

says, “You know if your demand in March was 58, you’d get a different answer.” Is

he right?Let’s use another Q&D and see.

A 0 1 2 3 4 5

B 35 45 58 77 120C 0 45 116 231

D 187 232 348 579

E 1 2 3 4

F 187 116 116 144.7

Yes indeed; the foreman saw the trigger point. Now let’s discuss how he did that.

Remember if the entry in the“F” row ever goes up,WE STOP,and order up TO that

point. Let’s look at the original problem again:

A 0 1 2 3 4 5

B 35 45 67 77 120C 0 45 134

D 187 232 366

E 1 2 3

F 187 116 122

Chapter 2


34/52


Look at the second entry in the “A” row (=1). Now divide that entry into the first entry in the “F” row (=187). This gives 187/1=187. This is telling you that you will

not order for the first period unless the demand in the second month is greater than

187. Say the February demand is 188, this gives:

A 0 1 2 3 4 5

B 35 188 67 77 120

C 0 188

D 187 375

E 1 2

F 187 187.5

As the cost goes up, but not by much, we would order for the first period only .

Following exactly the same logic and looking at the original problem again we

have:

A 0 1 2 3 4 5

B 35 45 67 77 120

C 0 45 134

D 187 232 366

E 1 2 3

F 187 116 122

We now do what the foreman did, and divide the third entry in the“A” row (=2),

into the second entry in the “F” row (=116). This gives 58, so he knows that you

wouldn’t order for just two months if the demand in March was 58 or less . The whole

idea of this method is, once you have fi lled in the forecast, the “bumps” in demand

should pretty well tell you where the break points are.As there may actually be times

when the big dogs need to know where the ball park is, we now display another

Silver and Meal Form that only requires the use of a straightedge. This is called the

Silver and Meal Nomograph and appears in the following section.

Chapter 2


35/52


Silver and Meal Nomograph ExampleSay that we have the problem given above, with an ordering cost of $566 and

holding cost of $3/item/period.On the nomograph form in Figure 11, we first find

on the left hand line, the value of our ordering cost of $566. Make a big black dot

there. Next, we find on the left-hand diagonal line, the holding cost of $3. Make

another big black dot there . Now lay a straightedge through these two dots and

make another big black dot on the index line in the middle of the page.

Now move to the right-hand side line and make a BBD at the “1,” to tell us that

we are considering ordering for one month. On the right-hand diagonal l ine find

the value of demand in January and make another BBD at 35. Now lay a straight-

edge through these two Big Black Dots, crossing the index line. If the straightedge

crosses the index lineabove it’s BBD,and it does, this tells you that you should con-tinue to the next period.

Moving again to the right-hand side line, make a BBD at the“2,” to tell us that we

are considering ordering for two months.On the right-hand side diagonal line make

another BBD at February’s demand of 45. Now lay the straightedge through these

two Big Black Dots, crossing the index line. Once more, the straightedge crosses the

index lineabove the BBD on the Index line. This tells us that we must look further.

Making a BBD at the third month on the right hand side line and also at 67 on

the demand line, the straightedge this time will pass below the BBD on the index

line.This tells us that it’smore expensive to order for three months than two,so our

fi rst order is 35 + 45 = 80 items to come in at the start of January.

CAUTION!

The nomogram or fill-in-the blank form above is the most accurate method of

Silver and Meal. The formula of:

T2*D(T) > S/CI

and the nomograph on which it is based are both very inaccurate,but easy to use

and easy to understand. The simple formula and the nomograph should never be

used for anything other than ball park estimates. You can, however, easily define a

nomograph to fit any situation where accuracy is not of great concern by using the

step-by-step method laid out in reference [6] at the end of this chapter.

Greening’s Nomograph for Varying Forecasted Demands

1.Locate your ordering cost/set-up cost on the far left line.

2.Locate your holding cost on the line just to left of center.

3.Using a straightedge,draw a line through these two points and through the index

line in the center.

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4.Locate the period you are in on the far right line.5.Locate your first period’s forecasted demand on the line just to the right of cen-

ter.

6.Using a straightedge,draw a line through these two points and through the index

line in the center.

7. If the new line crosses the index line aboveyour mark made in step three,

GO TO the next period and the next period’s demand and GO TO step 6.

8. If the new line crosses the index line below your mark made in step three,STOP!

Order the total demand up to (but NOT including) the period you are in.

Figure 11

Chapter 2

6

5

4

3

2

1

800

700

600

500

400

300

200

100

200

300

400

500

600

700

800

S /CI T2 D(T)

O R D E R I N G

C O S T S

P E R I O D S

MS

H O L D

I N G

C O S T S

( C I )

F O R E C

A S T D

E M A N D D ( T )

. 5

1

1 . 5

2

3

5

1 0

2

0

3 0 4 0

5 0 7 0

1 0 0 2 0 0


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When Do I Use Which Method?This is the end of commonsense methods of inventory control that I think are

worth presenting. I have repeatedly cautioned the reader about the technical

assumptions required for the methods, but another t ime never hurts.

EOQ

Only if the demand is constant, item cost is low,holding rate is reasonable,order-

ing cost is cheap and you took the above data yourself !


Remember, it’s really based on the EOQ.

Part Period Balancing

This requires that the holding cost line is equal to zero at an inventory level of

Q=0. This may not be true as your accounting system may have some amortized

fixed costs included. Check it out.

Dynamic Programming (Wagner-Whitin)

Only if your forecast doesn’t change and/or you are doing your inventory man-

agement on a Cray. If optimali ty is crucial, price is not a consideration and/or you

are the government, I suspect that Harvey Wagner will make it work.

Silver and Meal Should best be used on high value inventory with big set-up or ordering costs and

extreme variability in the forecast.

Final Suggestion

The RIGHT answer is probably none ofthe above, it’s most likely Just-in-Time,

(Lot Size = 1), with a KANBAN tracking system, if you can afford it and control

your forecast and your suppliers.Good luck!

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References

1.Harris,F. W.,“How Much Stock To Keep On Hand,” Factory, The Magazine of Management ,Vol. 10, No.3,pp.240-241, 281-284.

2.Kaimann,R.A., “EOQ vs. Dynamic Programming — Which One To Use For InventoryOrdering?” Producti on & Inventory Management , Fourth Quarter, 1969, pp.66-74.

3. Kaimann, R. A.,“A Comparison of the EOQ and Dynamic Programming Inventory ModelsWith Safety Stock Considerations,” Producti on & Inventory Management , Third Quarter, 1972,pp.72-91.

4.Silver, E.A.,and Meal, H.C.,“A Simple Modification of the EOQ for the Case of A Varying

Demand Rate,” Production & Inventory Management ,Vol.10,No.4,Fourth Quarter, 1969, pp.52-65.

5.Woolsey,R.E.D.,“Ordering Inventory When the Forecast Is Ridiculous,” Producti on & Inventory Management , Vol. 27, No.1,First Quarter,1986, pp.144-148.

6.Woolsey,R.E.D.,“A Nomograph For Ordering Inventory When The Forecast Is Ridiculous,”Production & Inventory Management , Vol. 27, No.4,Fourth Quarter, 1986, pp.128-133.

7.Woolsey,R.E.D.,“A Requiem For The EOQ,” Producti on & Inventory Management ,Vol. 28,No.3,Third Quarter,1988, pp.64-66.

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The Woolsey Never-FailSpare Parts Reduction

Method

If You’re Using the EOQ, At Least Do it Right!

In a previous chapter, I showed that continued use of the Economic Order

Quantity Model without knowing what it might be costing you was dangerous to

your continued corporate health. This chapter is a step-by-step example of check-ing the cost of inventory in a computerized spare parts inventory system still stu-

pidly using an Economic Order Quantity model to determine order levels and order

points.Following the method outlined below, almost anyone may be a hero to their

company or agency in a few short hours. All that is required is some slight knowl-

edge of the usual EOQ model, common sense, some basic programming ability and

a certain humorous level of entrepreneurship. A flowchart of the method is even

provided for academics with no practical experience of the workplace.

Setting the Scene

Once upon a time there was a nameless city with a Regional Transportation

Distr ict (bus company) that was being publicly pilloried by a local newspaper. It

seems that an investigative reporter had found some evidence that they were gross-

ly overstocked (or had grossly overordered) both spare parts and consumable

inventory.After the usual yelling and screaming (and resignations) had taken place,

things began to calm down somewhat.A local state university professor called when

things were at their most public and offered to come, at no charge, with students

and do an analysis of the computerized inventory system that had been installed

many years before. The RTD board responded quickly; clearly the prof and his

troops could do no harm, and the price was right.

With the Computer Jocks and What Happened There

The prof and his merry band hied themselves off to the RTD MIS group and

spent many happy hours learning the computerized inventory stocking and reorder

system.After demonstrating some competence, they casually inquired as to how the

reorder quantities were calculated. They received a classic explanation of the good

old reliable EOQ model and were shown a typical line for a stockkeeping unit that

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looked much like this:

PART # EOQ/ORDER POINT PRICE DEMAND OVER LAST # YEARS

CT1D4640 28/14 $4.44 287 3

Clearly, the item was ordered in lots of 28 whenever the on-hand fell to 14.

The ADP whiz was somewhat unnerved when the prof asked him to find the exact

statement in the program where the EOQ was calculated. A search of the program

(which happened to have been written in FORTRAN) revealed the following state-

ment:

C SET THE ECONOMIC ORDER QUANTITY102 EOQ= SQRT(2*D*R/C)

Here, (clearly) D is the annual demand in units.

C is the price of unit in dollars.

R is the ratio of ordering cost to holding cost.

As the demand and price are part dependent, we would really like to know what

the devil R is.

A search earlier in the program turned up the statements:

C SET RATIO OF ORDERING COST TO HOLDING COST

18 R = 12

But , if the program thinks that R = 12, Then this implies

R = (Ordering Cost)/(Holding Cost) = 12, which seems to imply that

Ordering Cost = 12 * Holding Cost.

At this point the prof asked what the analyst thought the holding rate really was

and received a most surprising answer. The civil servant said that, as it was govern-

ment money, it really didn’t cost anything to hold something in inventory. The prof

made a note of this and silkily asked him i f he might bring over a friend of his fromthe local newspaper that was giving them so much trouble (the city editor) so that

the civi l servant could repeat that statement for the press.At this point, this person’s

boss took this young man outside and screamed at him for a time. One supposes

that the boss explained to this budding civil servant the results of a headline that

said:

LOCAL RTD OFFICIAL SAYS TAXPAYER’S MONEYREALLYFREE

This would no doubt be followed by an announcement that this particular civil

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servant had just been sent to Great Sand Dunes National Monument to take inven-tory of the number of flies found on state issue flypaper,until a more worthy assign-

ment appeared.After this explanation, the offending civil servant, white-faced from

the previous discussion, asked humbly if the prof had any ideas for sett ing a hold-

ing rate.

The professor pointed out that as the RTD issued bonds,one had only to look at

the bond rating and a number could quickly be attached to the cost of money for

the RTD. This was done and found to be 11.4 percent. Plugging this interesting bit

of information into our formula above, we get:

Ordering Cost = 12 * Holding Rate = 12 * 0.114 = $1.37

The prof then asked the assembled multi tude if they believed that they could:

Note that they were out of stock.Do the paperwork for purchasing to launch the order.

Launch the order.

Receive the order.

Restock and update the record.

Pay the vendor.

And do all this for a grand total of $1.37.

At this point, the head warehouseman said that no way could it be done for less

than $20.But, if he thinks that then this implies:

Holding Rate = (Ordering Cost)/12 = 20/12

Thi s implies a holding rate of 166 percent a year . The assembled multitude round-

ly declared that both the ordering cost of $1.37 and the holding rate of 166 percentper year were manifestly absurd. We now have a dilemma; if an ordering cost of

$1.37 is absurd or the holding rate of 166 percent a year is absurd, then perhaps we

should have some doubts about R = 12.We have now been told that the ordering cost

must be at least $20, and the holding rate is 11.4 percent. Given these two bits of

information, we are forced to conclude that the FORTRAN statement that reads:

R= 12

Should perhaps really read:

R= 20/(0.114) = 175.43

Could it just be that this change will make a slight difference in our EOQ calcu-

lations?Could it also be that a consultant could make a profit here by offering to be

paid a (small) percentage of the savings?

The Long-Awaited Recalculation and What Happened Then

Let’s use the above knowledge to save some money.We fi rst find a typical item in

the spare parts list like the one we have seen before, namely:

PART # EOQ/ORDER POINT PRICE DEMAND OVER LAST # YEARS

CT1D4640 28/14 $4.44 287 3

It seems to say that a demand of 287 units was satisfied over the past three years

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by ordering in lots of 28 whenever an order point of 14 was reached. Now find theaverage annual cost of inventory for this item,using the present EOQ and the newly

extracted ordering cost and holding rate,using the tr ied and true formula for total

expected cost:

TEC = CIQ/2 + SD/Q, where: C is the price, I is the holding rate,

S is the ordering cost, and

D is the annual demand

We then plug in the price, the new holding rate, the new ordering cost, the old

annual demand, and the old “optimal” order quantity, which gives:

TEC = (4.44)(.114)(28)/2 + (20)(287/3)/28, or

TEC = Holding Cost(= $7.09) + Ordering Cost(= $68.33) = $75.42/year

Recall one of the many amusing assumptions made by the economic order quan-

tity model. At the optimal order quantity, the annual holding cost should equal the

annual orderi ng cost. Looking at the difference between $7.09 and $68.33,we imme-

diately realize that something is seriously amiss. The reason why this result is so

skewed is simple; the EOQ and order point that they have been using come from cost

data that probably hasn’t been updated since the system was installed . Using the well-

beloved formula for the optimal EOQ,

Now use this EOQ to get a different total cost:

TEC = (4.44)(.114)(87)/2 + (20)(287/3)/87, or

TEC = 22.02 + 21.99 = $44.01/year

The holding and ordering cost are now almost equal (as they should be.) The dif-

ference between doing it right and doing it wrong for this item alone is $31.41/year,

and this item is a relatively cheap one.

Chapter 3

EOQ =2*S*DC*I

=2(20)(287/3)(4.44)(.114) = 87 units


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Icing on the Cake — Recalculating the Order PointWe recall that the above part had an order point of 14. Order point is usually

defined as

OP= (Annual demand)(Lead time in days)/365

Knowing that the cost data hadn’t been updated since the system was put in,

might one just suspect that the lead times presently i n the program probably go back to

the same date . Some careful questioning revealed that while they used to get their

bus parts from Peoria, Ill ., they were now getting most of them from a bus manu-

facturer about three hours away. In short, the new order point was now zero for

about 95 percent of their parts. In short, the old order point was EOQ/OP = 28/14,

and the new one is: EOQ/OP = 87/0.All that now remains is to write a Mickey Mouse program to:

Read in the spare parts history file for each inventory item.

Calculate the correct EOQ and order point.

Find the difference in cost between the new and old EOQ.

Sum up the total expected cost differences for all of the items.

Print the total amount that will be saved with the new EOQs.

This type of analysis at a nameless coal mine in Kentucky showed that over 30

percent of an inventory of 36,000 SKUs (stockkeeping units) would have a new

EOQ/OP of 0/0. In short, over 30 percent of the multi-million dollar inventory

should never have been stocked at all . The application of this method to a nameless

Canadian auto parts firm resulted in a savings of over $2.4 mil lion dollars in the

first year. It is rumored that some nameless consultant was granted (contractually)

a teensy percentage of the savings.

Final Warnings and Suggestions

It is exceedingly important that the casual reader be explicitly warned against

indiscriminate use of the method outlined above.First, in all of my consult ing work

in a number of countries, I have yet to see one single example of an EOQ model

being applied where all of the required assumptions for its use were met. In the

above example, we have generated considerable (apparent) savings by simply updati ng

a model . A much more important question to ask is whether or not the EOQ is the

right model for this inventory system at all (it probably isn’t). The best use for the

method here is to generate enough (initial) savings to get management’s attention.

If you save them a few million, they might even let you build a system to do it right .

To make this process completely explicit, all that is required is to follow the flow-

chart given at the end of this chapter, usable in both the pubic and private sector.

Mili tary and/or government employees who try this, while they are not allowed to

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Chapter 3

get a piece of the action, can at least get a promotion and/or a medal out of it. Inconclusion, for those of you in both the public and private sectors I can only say,

good luck and good hunt ing .

References1.Woolsey, R.E.D., and Swanson, Huntington S.,Operations Research For Immediate Application, A Quick & Dir ty Manual , Harper & Row, New York, (1975), pp.39-41.

The above article originally appeared as:

2.Woolsey, R.E.D.,“The Never-Fail Spare Parts Reduction Method: An Editorial,” Production and Inventory Management Journal ,Vol. 29, No. 4, Fourth Qtr.1988, pp. 64-66.


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Flowchart of the Woolsey Never-Fail Spare Parts Reduction Method

Step 1. Find a computerized spare parts inventory system stil l

using the EOQ.

Step 2. If cost updated recently,GO TO Step 1. If not,GO TO

Step 3.

Step 3. Use trapping method to find out holding rate and set-

up or ordering cost.

Step 4. If costs are not absurd, GO TO Step 1. If costs areabsurd, GO TO Step 5.

Step 5. Using real costs, find total annual cost of ordering

spares using present operating method.

Step 6. Also using real costs, calculate right EOQ and order

point. Then, calculate annual cost of inventory under

right operating conditions.

Step 7. Calculate difference in annual costs for each spare

part between new and old ordering policies.

Step 8. Sum up cost differences for total inventory.

Step 9. Obtain (contractually) fee of 1 percent of above dif-

ference before telling them magnitude of saving.

Step 10. Tell them what they owe you, get money.

Step 11. Reside in the Bahamas.

Ste

Documents

Inventory Control by Woolsey and Maurer