3,1

Reprinted fromECONOUKTXXCA, Journal of the Econometric Society, Vol. 20, No. 2, April, 1952

The University of Chicago, Chicago 37, Illinois, U.S.A. Printtd in U.S.A.

ON THE APPLICATION OF SERVOMECHANISM THEORY

IN THE STUDY OF PRODUCTION CONTROL

BY HERBERT A. SiMON1

The problem of controlling the rate of production of a single product

can be stated in terms of servomechanism theory, and the well-developed

methods of that theory employed to study the behavior of a control sys­

tem. This is illustrated for a simple system, and a cost criterion is con­

structed for evaluating alternative decision rules or constructing an

optimal rule. Laplace transform methods are introduced, and some of

their elementary uses for studying the stability and steady-state behav­

ior of systems are illustrated.

1. INTRODUCTION

THIS PAPER is of an exploratory character. Powerful, and extremely

general, techniques have been developed in the past decade for the

analysis of electrical and mechanical control systems and servomecha-

nisms. There are obvious analogies between such systems and the human

systems, usually called production control systems, that are used to

plan and schedule production hi business concerns. The depth or super­

ficiality of these analogies can be tested by subjecting a fairly simple,

but relatively concrete, example of a production control system to some

of the techniques of analysis usually employed for servomechanisms.

No attempt will be made, hi this introductory essay, to do justice to

the full range of analytic tools available to the servomechanism engi­

neer for synthesizing control systems. Our intent is to give an ele­

mentary introduction to servomechanism theory and to determine its

applicability to production control problems.It might be pointed out that the notion of a servomechanism incor­

porating human links is by no means novel. In particular, many gun-

sighting servos involve such a link. The idea of social, as distinguished

from purely physiological, links is relatively new. However, Richard

M. Goodwin [12] has arrived independently at the same idea as a means

for studying market behavior and business cycles. The applicability of

servomechanism models to the theory of the firm has been discussed by

my colleague, W. W. Cooper [8] (see references at the end of paper).

Two dynamic macrosystems that have been represented by analogue

circuits (and hi one case experimentally investigated) may also be re-

1 1 am indebted to W, W. Cooper, C. Klahr, and David Rosenblatt, and to staff

members of the Cowles Commission for numerous helpful suggestions. The re­

search was undertaken in my capacity as a consultant to the Cowles Commission

for Research in Economics under its contract with the RAND Corporation. This

paper will be reprinted as Cowles Commission Paper, New Series, No. 59.

247

248 H. A. SIMON

garded as servomechanisms [9, 10]. All such systems would be included

in Wiener's [11] general program for cybernetics.Some preliminary remarks are necessary, first to characterize servo-

mechanism theory, and second, to describe the production control sys­

tem we will study.Servomechanism theory. Many of the systems with which electrical

and mechanical engineering deal are described, at least approximately,

by systems of linear differential or integro-differential equations with

constant coefficients. Included are electrical networks with lumped con­

stants, and among these (or their mechanical analogues) are many of

the systems known as "controllers," "regulators," and "servomecha­

nisms." We will not attempt here to distinguish these terms precisely,

but instead set forth an example of such a system.Consider a system consisting of a house, or other enclosed space, a

gas furnace, and a thermostat that controls the rate of gas flow in the

furnace.2 The temperature setting of the thermostat (that is, the desired

house temperature) will be referred to as the input of the system, desig-

e,y.» \K-> >* K, *

£• ^ A 'T *<t e,

W

FIGURE 1

nated by 0/ ; the actual house temperature will be referred to as the

oviput, 00 ; their difference, 0/ 80 , as the error, e; and the outside

temperature as the load, 6L .* All variables are functions of tune.The system is so constructed that the rate of gas flow in the furnace,

and hence the rate at which heat is supplied to the house, is a function

of the error (in the simplest case, proportional to the error). Further,

this function relating the error to the output is selected so that the

error will tend to be reduced, whatever the load imposed on the system.

8 In order to preserve as close an analogy as possible with the production con­

trol system to be described later, we will assume a thermostat that is continuous

in operation, instead of the more familiar on-off thermostat. The system described

here is analysed in reference [4, pp. 298-303].1 In our example, the input is generally fixed, while the load is variable. This

is the typical case of the controller; and the input is frequently referred to as

the standard. The term "servomechanism" more commonly refers to a system in

which the input is variable and the variable load absent. However, in many im­

portant engineering systems both variable input and variable load are present.

O

o

•oSBRVOMECHANISM THEORY IN PRODUCTION CONTRO

L 249

The system just described is shown in Figure 1. The equations that

describe the system are

. at

(1.2) e = 0, - e0 .

The symbols © indicate differential devices (subtraction); the box

Kz corresponds to /(e); and the box Ki to the integration [from equa­

tion (1.1)1 which gives 00 as a function of /(e) and (Q0 — OL).Two important features of this system should be noted. The first is

the control loop, or feedback loop (the upper loop in Figure 1), by means

of which (a) the output is compared with the input and (b) their differ­

ence is fed back into the system to alter the output in the direction of

reducing the difference.The second important feature is shown by the directional arrows.

The input and load affect the behavior of the system (and, in particular,

the output and error) but are themselves unaffected by it. Hence, varia­

bles not included in the loop may be regarded as independent and may

be assumed to have any arbitrary time paths. This kind of relationship

is sometimes referred to as unilateral coupling, or cascading. A reciprocal

relation must be represented by a closed loop (such as the lower loop

in Figure 1).

nln a physical servomechanism, cascading is made possible by the

fact that the closed portion of the system involves very little energy

in comparison with the energy of the independent variables (as in a

solar system with large central sun and small planets) or, more generally,

draws its energy from an independent power source (as in an amplifier).

It is this characteristic of the system that permits the output to follow

the input without disturbing the path of the input. A servomechanism,

then, is a system (1) unilaterally coupled to an input and a load, (2)

with one or more feedback loops whereby the output is compared with

the input, and (3) with a source of energy controlled by the error that

tends to bring the output in line with the input. If the load is bilaterally

coupled with the output, then the former must be included in the sys­

tem and cannot be treated as an independent variable.

The most powerful technique for treating servomechanisms employs

the Laplace transform. (See [2, Chapter 2; 31). The Laplace transform

of the input may be interpreted as its decomposition into its component

frequencies (i.e., it is very closely related to the Fourier integral).

* The Laplace transform of the entire servosystem connecting input with

output describes its behavior in filtering (altering amplitude and phase)

the frequencies occurring in the input. The Laplace transform of the

n

250 H. A. SIMON ,,

output, which is the product of the two previous transforms, is the representation of the output in terms of component frequencies. (In this statement we disregard the load, which enters as another input.) The system is studied by determining the Laplace transform of the servo, multiplying this by various input transforms, and analyzing the resulting kinds of behavior of the output. We are interested hi the stability of the output (which is related to its transient response) and in its steady-state behavior for various inputs. By defining a criterion function (a function of the output) we may compare the merits of alternative servos for controlling an output under specified conditions.

Production control. In this paper we shall consider the control of the rate of production of a single item. The item is supposed to be manu­ factured to standard specifications, placed in stock, and shipped out on order of customers. The item is manufactured continuously, and control consists in issuing instructions that vary continuously the quan­ tity to be manufactured per day (or other unit of time).

The aim of the control system is to minimize the cost of manufacture ^^^^

over a period of time. This cost, or the variable part of it, is assumed )

to depend upon (1) the variations in the manufacturing rate (i.e., it costs more to make 1,000 items if the manufacturing rate fluctuates than if it is constant); and (2) the inventory of finished goods (i.e., increase in this inventory involves carrying costs, decrease in the in­ ventory below a certain point involves delay in filling customers' orders). Hence, the criterion by which we will judge the system will be some function of the magnitudes of the fluctuations in manufacturing rate and the inventory of finished goods.

We will take as our input the optimum inventory (0,). Since this will be assumed constant throughout our problem, it may be taken as zero. The actual inventory of finished goods will be taken as the out­ put (0o)-4 The error (c) will then be the deficiency (positive or negative) of inventory (0/ 00). Customers' orders per unit of time will be treated as the load (0L). We need two additional variables, the actual produc­ tion rate per unit of tune (/*), and the rate of planned production or, more accurately, planned new production per unit of tune (17).

Assume that, on the basis of information about orders and the in­ ventory excess or deficiency, instructions are issued daily (in our model, continuously) for the manufacturing rate of the product of a certain number of units. At some later date, the lag being determined by the

4 The actual inventory may be either positive or negative. A negative inven­ tory is simply a backlog of customers' orders. Wherever the term "inventory" ,

is used in this paper it should be interpreted as "inventory of finished product or backlog of orders." Depending on the commodity to be produced, the optimum inventory may also be either positive or negative. The former is the case in a ^^^

plant that ships from stock; the latter, in a plant that manufactures on order. j

y-x.

8ERVOMECHANI8M THEORY IN PRODUCTION CONTROL 251

time required for production, the units of product put into production

at the initial time are actually produced and added to inventory. Mean­

while, customers' orders have been daily (continuously) withdrawn from

inventory. Information regarding the inventory level is in turn fed

back to be compared daily (continuously) with the optimum inventory,

and the calculated error employed, in turn, to redetermine the planned

production rate.This system obviously possesses the characteristics of a servomecha-

nism. It is unilaterally coupled to the load and input (customers' orders

and optimum inventory). It has a feedback loop: error * planned pro­

duction -* actual production * inventory * error. The error initiates

a change in planned production hi such a direction as to reduce the

error.In succeeding sections systems will be described for accomplishing

the functions just listed. We will start with some highly simplified struc­

tures, and add complications as we proceed.

eo1 e i« - — \ K 2

1

t«L

FIGURE 2

k

2. A SIMPLE SYSTEM FOR INVENTORY CONTROL

We will consider two systems. In the first we will be concerned only

with the control of inventories; we will base production decisions only

on information about inventories (and will ignore information about

orders); and we will assume a zero tune lag for production. In the second

system all these restrictions will be removed.Description of the system. The first system is shown in Figure 2, which

is identical with Figure 1 except for the absence of the lower loop that

appears hi the former. In this system, it is assumed that n is identically

equal to tj. That is, the rate of production at tune t is equal to the rate

at which new production is scheduled at tune t. This implies that pro­

duction plans are carried out without an appreciable tune lag. The equa­

tions of the system are

(2.1)

(2.2)

KI and Kz are linear operators whose form will be specified. Equation

(2.3) is a definition. Equation (2.2) represents a rule of decision it

252 H. A. SIMON

specifies the rate of production that will be scheduled (and achieved)

as a function of the excess and deficiency of inventory. The precise

form of (2.1) is determined by the conditions of the problem since, by

definition,

(2.4) *s = M - eL.at

Hence, if we wish to design a servomechanism of the class described

that meets some criterion of optimality, we have at our disposal only

the operator Kz—the decision rule.Our equations (2.1)-(2.4) can be restated in terms of the Laplace

transforms of the quantities involved. The real variable t is replaced

by the complex variable p. The Laplace transform of y(i), written

y (P), is defined by

(2.7)

(2.8) M(P) =

(2.9) «(p) - et(p) - B0(p\

(2.5) y*(p) = I y(f)e~pt dt.

This integral exists for a wide class of functions, although in some ,!/ J

cases it must be defined as a Lebesgue uitegral rather than a Riemann

integral. The inverse transformation is

(2.6)

where the path of integration runs parallel to the imaginary axis along

the line: real part of p = 6.6It can be shown that the Laplace transform of the derivative of a

function that is initially zero is p tunes the transform of the function.

Using this relationship, we can transform the terms of (2.4) and obtain

(2.7), below. Comparing (2.7) with (2.1), we see that the operator KI

in the t domain (integration) corresponds, hi the p domain, to multipli­

cation by 1/p. In equations (2.8) and (2.9) we write the relations in the

p domain that are obtained by transforming (2.2) and (2.3), respectively.

We then have

where 0o(p), etc., represent the Laplace transforms of 0o(0» etc., re­

spectively. [Since the argument of each function indicates whether it

' No attempt will be made in this paper at mathematical rigor. Virtually all

the mathematical tools employed here will be found in [2, Chapter 11] and [8]. The

latter also contains (pp. 332-357) a very useful table of Laplace transform pairs. X*N

n8ERVOMECHANI8M THEORY IN PRODUCTION CONTROL 253

is defined in the p domain or the I domain, in these equations and those

following we omit the asterisk (*) in writing a transform.] If we now

assume 0/ = 0 and introduce the system transform,

(2.10) y(p) -gg,we derive from (2.7H2.9)

(2.11) Y(p) = t^

Theorems about the Laplace transform. The behavior of the system

under varying load can be discussed in terms of the properties of the

system transform, Y(p). As a basis for this discussion we will outline

some results from Laplace transform theory without attempting proofs

or complete rigor in their statement.A system will be termed stable if the output remains bounded (in

the t domain) for all bounded inputs (in the t domain). The equation

obtained by setting the denominator of 7(p) equal to zero we call the

characteristic equation of F(p). In our particular system we have

(2.12) p + Ki(p) - 0.

Provided that the numerator of F(p) has no finite poles, the system

will be stable if and only if all the roots of the characteristic equation

iT-v have negative real parts.* ' ' Let W (0 be the inverse transform of 7(p), as defined in (2.6). We

call W(t) the weighting function of the system. Multiplication in the p

domain corresponds to convolution in the t domain. Hence, we have,

from (2.10),

(2.13) 60(t) = W(r)eL (t - T) dr.

Equation (2.13) relates the time path of the output to the weighting

function of the system and the time path of the load. Generally, however,

we do not employ this relationship. Instead, we multiply the Laplace

transform of 6L(f) by the system transform and then take the inverse

transform of this product, obtaining 0(0 directly. Indeed, it is this

procedure, together with the availability of tables of transform pairs,

that makes the Laplace transform method particularly powerful.

Two additional theorems, which hold when the indicated limits exist,

will prove useful:

(2.14) Lim y(t) = Lim py(p),|-.oo

(2.15) Lim y(t) - Lim py(p).*-»o

254 H. A. SIMON

In particular, (2.14) enables us to calculate immediately the steady state output for given load and system transform without transforming again to the t domain.

Steady-state and transient behavior. We return now to the task of dis­ covering for our particular system a transform, Kt(p), that will induce appropriate behavior of 60(f). By "appropriate" behavior we mean that we wish Q0(t) to be as small as possible. We consider first the steady- state behavior, which we will study by means of (2.14).

(2.16) Lim00(# = IAmp60(p) = Lim -&

A. Suppose that up to the time t = 0, orders have been zero, and that after that time they are received at the rate of 1 order per unit of time,(2.17) 6L(t) = 0 for t < 0; 6L(t) = 1 for t% 0.

We have

(2.18) £[6L(t)] = 0L(p) = l/p. Hence,

(2.19) Lim 8o(t) = Lim -p

Therefore, we wish the denominator of the right-hand side of (2.19) to become very large as p approaches zero. This can be accomplished, for example, by setting

(2.20) #2 = (l/p*) (a + 6p) with k ^ 1, a > 0, 6 > 0.

Rapid convergence is assured by making a large.B. Suppose, now, that up to the tune t = 0 orders have been zero,

and that after that time they are received at the rate of tn orders per unit of time,

(2.21) 0L(i) = 0 for t < 0; 0L(t) = tn for t > 0.

We have

(2.22) £[«*«)] =

In this case we can assure a zero steady-state error with Kt of the same form as (2.20) but with k ^ (n + 1).

C. Suppose that 6L(f) is sinusoidal:

(2.21) 8L(t) = 0 for t < 0,

(2.22) 0L(t) = A/2[eM + e~*"] = A cos at, for * £ 0.1

8EBVOMECHANISM THEORY IN PRODUCTION CONTROL 255

Here we cannot use the method of the previous two cases since it can

be shown that Limt_»oo 60(t) does not exist. Instead, we use the result that

if 0L(£) is sinusoidal, 60(f) will be sinusoidal (aside from the transient term),

with the same frequency but altered amplitude and phase. That is,

(2.23) Lim0o(*) = B cos (ut + $.i-»oo

The amplitude, B, of the output is given by

(2.24) B = A[Y(iu)Y(-iu}f.

If, for example, K* = (a/p) 4- 6, we have

(2.25) 7(p) = p + Ka(p) p2 + bp + a (p_ pl)(p -Pa)'

where pi and pj are the roots of the characteristic equation.

( • tco —

(2.26) w2

(2.27) J5

For given pi , pa , as w approaches zero, B approaches zero; as w grows

large, B approaches A/w. When pi and pa are equal, B approaches its

maximum f or w = pi . This maximum is B — A/2pi . Hence we see

that by selecting K3 so that the characteristic function has large roots

we guarantee rapid damping of 60 for sinusoidal loads.6

We have now indicated the properties that our decision rule (the

operator K2) must possess to assure small or vanishing steady-state

inventory excesses and deficiencies for various loads.

Next, let us interpret these results in the t domain. Suppose that

y(p) is an algebraic expression,

(2.28) y(p) =

' In analogy with the fact that a wide class of functions can be represented

by Fourier sums of sinusoidal functions, the steady-state analysis of arbitrary

loads can often be handled in servomechanism theory by decomposing the func­

tion representing the load into a weighted integral of sinusoidal leads with con­

tinuously varying frequencies. For this reason, the restriction of the load in the

steady-state analysis of subsequent sections of the paper to a simple sinusoid

does not involve any essential loss of generality with respect to the form of the

function representing customers' orders. See, however, footnote 8.

an p" + an-ip*-1 + 4-ao

r

256 H. A. SIMON r^

Then the equation in the t domain obtained by transforming (2.10) is d"00

(2.29)W"fl_

&m~l

If, for example, 7(p) is defined by (2.25), we have

(2.30) . at atFrom this equation we can verify the conclusions already reached

above. For example, hi A, we had d6L/dt = 0 for t ^ 0. Here the general solution of (2.30) is

(2.31) 80 = Me>lt + Neptt,

where p\ and pa are the roots of the characteristic equation,

(2.32) p = -b±Vbr=~&,Zi

Since a > 0, 6 > 0, pi and p2 will be real and negative or complex with negative real part, and hence (2.31) will converge to zero as t increases.

If, on the contrary, we had deL/dt = 1 for t ^ 0, the general solution of (2.30) would be

(2.33) 80 = Mflt + N<?*' + (I/a).Then the system transform (2.25) would yield a steady-state error

of I/a as t —» oo. Again this result can be obtained directly by sub­ stitution of (2.25) and 0L(p) = 1/p2 in (2.16). Moreover the transient part of 80 will be rapidly damped if the negative real parts of pi and Pt are large.

Returning to the case of a sinusoidal load,

(2.34) 0L (p) = £[cos <*t] = -T-4--,p -p w

with the system function of (2.25), we have

(P ~

A B Cp + D(P - Pi) (P - P*) P2 + wj '

The transform of this is

(2.36) e0(t) = A<?lt + Bf*' + E cos

SERVOMECHAN1SM THEORY IN PRODUCTION CONTROL 257

The final term of (2.36) we have already encountered in (2.23) it

is the steady-state response to the sinusoidal load. The first two terms

represent the transient response which, again, will be rapidly damped

if pi and p* have large negative real parts.Stability of the system. We stated in the section before last that a sys­

tem will be stable if the roots of the characteristic equation of the system

transform have negative real parts. In the section just preceding we

noticed that the transient response of the system is independent of the

load and is determined by the roots of the characteristic equation. If

the roots have large negative real parts, the transient will be strongly

damped. These results suggest that many properties of the system can be

determined directly by examination of the roots of the characteristic

equation. We next carry out this program for various choices of Kz .'A. Let KZ — a/p with a real. Then po = ±\/a» an(i the system is

unstable since at least one of the roots has a nonnegative real part.

®' L^ ^2 = a/P + &> with ° and & real. Then p0 =

v s. ( b ±\/b2 4a)/2, and the system is stable if a > 0, b > 0; other-

wise, unstable. This result has already appeared from (2.31) and (2.32).

C. Let K* = a/p -f- 6 + cp, with a, b, c real. Then po =

[ 6 ± V&2 4a(c + l)l/2(c + 1), and the system is stable if o, 6,

and (c 4- 1) all have the same sign; otherwise, unstable.

Interpretation of the decision operator. The operator Kz represents a

rule of decision. Since n(p) = K2(p)-e(p), this rule determines, on

(l the basis of information as to the current deficit or excess of inventory

[«(p)]> a* what rate [n(p)] manufacture should be carried on. Among the

operators that have been examined and found to possess satisfactory

properties is K2 = a/p + &, with a and b large positive constants. With

this operator, equation (2.2) becomes

(2.37) = <««) + 6 ,

which, interpreted, means: the rate of production should be increased

or decreased by an amount proportional to the deficiency or excess of

7 We will not employ in this paper some of the procedures, such as Nyquist's

rule, widely used in servomechanism analysis to determine whether a system has

any roots with positive real parts. For Nyquist's rule, see [2, pp. 67-75] and [5,

Chapter V]. It may be appropriate at this point to emphasize the exploratory

intent of this paper. Emphasis has been placed on formulating the problem in the

language of servomechanism theory, determining the criteria for evaluating the

merit of a control system, and surveying the general basis in servomechanism

theory for approaching such problems. For a more adequate account of the wide

collection of analytic and graphical tools at the disposal of the servomechantsm

engineer for synthesizing a control element that has the desired characteristics,

/~*\ the reader must turn to the references at the end of the paper.

258 H. A. SIMON )

inventory plus an amount proportional to the rate at which the inven­

tory is decreasing. The constants of proportionality, a and 6, should be

large if it is desired to keep the inventory within narrow bounds. The

relation b2 > 4o should be preserved if oscillation is to be avoided.

All of this is obvious to common sense. What is perhaps not obvious

is that derivative control [e.g., the final term in (2.37)] is essential to

the stability of the system. To base changes in production rate only

upon the size of inventory (setting 6 = 0) would introduce undamped

fluctuations in the system. (Compare A and B of the previous section.)

3. SYSTEM WITH PRODUCTION LAG

With this preliminary analysis of a simple system we are ready to

study a system that approximates more closely to the problems we would

expect to encounter hi actual situations. The most important features

missing from the previous system are a production lag and the availa­

bility of information about new orders. In actual cases a period of time

will elapse from the moment when instructions are issued to increase

>*0

FlOTJBE 3

the rate of production to the moment when the increased flow of goods

is actually produced.Description of the system. In Figure 3 is shown a system with a pro

duction lag. The equations of this system are

(3.1) 60 = Kfo - *!,),

(3.2) M =X4n,

(3.3) r, =K# + K>8L ,

(3.4) € = 61 - Bo .

The new variable q(Q represents the instructions at tune t as to the

production rate; p(f) now represents the actual rate of production of

finished goods at time t. p(p) and ij(p) are connected (3.2) by the "pro­

duction lag operator," K* . As before, we will have, by definition, K\ — 1/p. The operators Ka and Kt correspond to the decision rule, which now

depends both on inventory level and rate of new orders. Both operators

are at our disposal in seeking an optimal scheduling rule. It remains to

find a plausible form for X4 . / -N

SERVOMECHANISM THEORY IN PRODUCTION CONTROL 259

The simplest assumption is that the production lag is a fixed time

period,T. That is,

(3.5) M(0 = T,(* - r).

This means that if a given rate of production is decided upon at a

certain time, this rate of production of finished items will be realized

r units of time later. The operator transform K^(p) corresponding to

(3.5) is

(3.6) K*(p) = e-*.

Substituting the known functions Ki(p) and K^(p) in the system equa­

tions and solving for the system transform, we get

- 6o(p} - (e~T*K* ~ l} ~ -

A comparison of (3.7) with (2.11) reveals that both numerator and

denominator have been affected by the introduction of the production

lag. Hence we shall have to re-examine the entire situation.

Feedforward of information about new orders. Consideration of the

numerator of (3.7) shows that the control of inventory is not a trivial

problem. If we set K* = 1, the numerator becomes (e"1* 1), which

approaches zero only as p approaches 2mrt/T, where n is zero or an

integer. Hence, this procedure would stabilize the inventory perfectly

_^ only for a sinusoidal load whose frequency is an exact multiple of the

I frequency corresponding to the production lag. At best we can say that

the system will perform better with the operator KI * 1 than without

any information about orders, but by no means perfectly.

Why not set Ka = erp ? Then we would have (e^Ka - 1) = 0. Define

the variable <f> so that

(3.8)

Taking the inverse transforms of both sides, we find

(3.9) *(0 - 6L(t + r).

Hence, setting X8 = erp corresponds to predicting the value of QL

for T units of tune in advance of the actual receipt of orders. Again the

result is intuitively obvious. If we could predict orders over the time

interval T, we could schedule production in anticipation of the actual

receipt of these orders and avoid any inventory fluctuation whatsoever.

We will not explore further the problem of forecasting 6L(t + T}, but

will consider optimal decision rules when future orders are not known

with certainty.8

* In work that is continuing on this problem in an Air Forces project at Carne-

gie Institute of Technology, an attempt is being made to reformulate the problem

260 H. A. SIMON

Feedback of information about inventories. We consider next the denominator of (3.7). Because of the sinusoidal character of e"**, this will behave roughly like (p -f- X2). Hence, the system will behave in the same general manner as the system analyzed in the second section. Moreover, because of equation (2.14) we may expect the same general behavior in the steady state of this system as of the system of Section 2.The roots of the characteristic equation(3.10) p2 4- (a + bp)e~*p = 0are not easily evaluated. We will not investigate further here the roots of this transcendental equation. Instead, we will suggest a method of replacing the fixed lag, with operator e"**, by a distributed lag, with operator a2/(a2 + p2), which retains the algebraic character of the system transform and avoids the difficulties encountered in handling(3.10).

In place of n(t) = ij(t — r) we write

(3.11) M(0 = / P(r}rj(t - r} dr, Of where

P(T) dr = 1.

P(r) may be regarded, then, as the probability that the lag in pro­ ducing a particular scheduled item will be of length T. For large values of T we would expect P(r) to be zero, or at least very small. If (3.11) holds, we have, by (2.13),(3.12) n(p) = P(p)i;(p).

For example, suppose P(r) — a*re~aT. Then P(p) = a2/(a2 + p2), and( ' *-\a2 4- rr(3.13)

a

If we define f = / rP(r) dr — 2/a as the mean lag, we see that the mean lag is still independent of 17. The system transform defined byin stochastic terms. In this approach the customer order function is regarded as an autocorrelated function rather than a sum of superimposed sinusoidal func­ tions. When the problem is looked at in this way, the rather artificial distinction we have maintained between the prediction problem and the filtering problem tends to disappear. The stochastic approach, as applied to servomechanism theory, is largely the work of Norbert Wiener. See References [6], and [2, Chap­ ters VI-VTII]. Our work has not yet progressed far enough to indicate the range of usefulness of the stochastic methods in relation to problems of the kind con- sidered here.

V: 8ERVOMECHANISM THEORY IN PRODUCTION CONTROL 261

(3.13) can be analyzed by the methods previously employed to deter­

mine suitable forms for K3 and Ks. For example, if we take K3 = 1, K2 = (b -f- cp2),

(3.14) 7(p) = s 2"? a2p + 026 -

This has zero steady-state error for 6L = 1/p2, i.e., for 8L = t (t ^ 0). \

The parameters b and c can now be given such values that the real parts !

of the roots of the characteristic equation will be negative, and the j

system consequently stable. The necessary and sufficient conditions

for this are c > b > 0. \

4. CONTROL OF INVENTORIES AND PRODUCTION-RATE FLUCTUATIONS :

The general criterion for the optimality of a production control |

system of the sort we are analyzing is that cost of production, in some I

sense, be minunized. |

«>s Large inventories involve interest costs, possible costs through physical

'. depreciation in storage, warehousing costs, etc. An inventory deficiency j

(excessive negative inventory), on the other hand, involves a "cost" [

in the sense of delay in filling orders, and consequent customer ill will. I

It appears reasonable to include in the cost of production, therefore, I

an element that represents the cost of excess or deficiency in inventories, |

say £i(0o). In first approximation we may take & proportional to | 60 |, I

or to 0o- 1

It also appears reasonable to assume that the cost of producing a given j

quantity of output over a period of tune is minunized if output is con- j

stant during that tune. If we represent the output as a constant plus j

an oscillating function with zero mean n(t) = M + n(t)—then we ;

may assume that the rate at which cost is being incurred is a function j

of M and of the frequency and amplitude of /*(<). j

Now, from equation (3.1), we know that \

(4.1) t

that is,

(4.2) at

Hence, if we succeed in stabilizing 00 at B0 = 0, M will not be con­

stant but will follow 6L(f). Conversely, if we stabilize M> Qo will not be

constant, but will follow the integral of 6L(t). We cannot devise a sys­

tem that will simultaneously eliminate inventory and production ftuctua-

^ tions, but must, instead, establish a criterion that is some weighted

fi average of these.

262 H. A. SIMON \J

Analysis of a specific criterion. To be specific, we consider the steady state of the system under sinusoidal inputs and outputs. This assump­ tion is consistent with the system (4.2). In fact, in the steady state, if BL is sinusoidal, B0 and n will be sinusoidal with the same period. We assume that the cost associated with /* is proportional to the square of the amplitude of its oscillation, i.e., that it is of the form p | B | 2, where | B | is the amplitude. Similarly, we assume that the cost of holding inventories is <r \ C j 2, where | C | is the amplitude of 80 .

We let

(4.3) 8L(f) - a cos ut,

(4.4) A*(0 & cos dit + ft sin tat,

(4.5) 0o(0 = c cos ut -f 7 sin at, with a, 6, ft, c, 7 real, whence

(4.6) «7 = b - a, -we = 0.

We wish now to minimize

(4.7) p(&2 + (?) + <r(c2 + 7*) = £

subject to (4.6). Substituting for c and 7 from (4.6) into (4.7), taking derivatives of £ with respect to 6 and j8, and setting these equal to zero, we find

(49) c = 0, 7 <**"+ •0

For small «: 6 > a, 7 > 0. For large w: 6 » 0,7 > 0, &ry > o.Interpreting these results, we find that the optimal decision-rule will

adjust the production rate and hold inventories down for long-period fluctuations in orders, but will stabilize production and permit inven­ tories to fluctuate for rapid fluctuations in orders. In the latter case, the inventory excess or deficiency will remain small (7 > 0) because the period of oscillation is short. The amplitude of manufacturing fluctuations (6) will vary inversely with to. The magnitude of inventory fluctuations (7) will have a maximum for w2 = o/p.

An alternative criterion. In the previous section we used the quadratic cost function (4.7). Interesting results are obtained by using the linear function,

(4.10) f = p

o8ERVOMECHANI8M THEORY IN PRODUCTION C

ONTROL 263

Minimizing f after substitution for c and 7 from (4.6), we find as

optimum values

(4.11) ft - 0, c = 0.

But for b we find

(4.12) b = 0 for co > <7/p, b = a for w < o/p.

Correspondingly, for 7,

(4.13) 7 = a for « > o/p, 7 = 0 for w < o/p.

Writing Z(p) = n(p)/6i(p), we see that for optimum results, Z(p)

should have the characteristics of an ideal low-pass filter: it should

transmit without distortion all frequencies below 2ira/p and should filter

out all frequencies above 2ir0/p. The meaning of this requirement hi

terms of a decision rule can be interpreted by the same methods as

those used for the quadratic cost functions in succeeding sections.

Requirements for the system transform. Returning to the quadratic

cost function of Section 4, we must now determine what kind of a system

transform will satisfy (4.4)-(4.5) with 6, /3, c, 7 given by (4.8)-(4.9).

n For brevity we write 00(p)/0z,(p) = Y(p), n(p)/0L(p) = Z(p). Then,

from (4.1),

(4.14) Z(p) = 1 + p7(p).

The optimum transform Z(p) is found readily as follows. Recalling

(2.13) we can write the output /*(<) for a sinusoidal load eM\r"° f*

(4.15) M(<) - / Z(r)e<"( '~T) dr = e*"' I Z(r)e~^ dr.Jo Jo

But the factor under the integral sign of the right-hand side of (4.15)

is, by definition, Z(tw). That is, for a sinusoidal load with period 2»ci>,

we will have

(4.16) M(«) = Z(i«)0i(0.

We see immediately that

(4.17) z(ico)=> = _^ .o par -f- <r

Hence,

and, from (4.9),

(4.19) Y(p) = - pp

o264 H. A. SIMON

The characteristic equations of Z(p) and Y(p), however, have real roots of opposite signs, p = ±(<r/p) . Hence a system with these trans­ forms would be unstable. The transient output would increase ex­ ponentially.

The reason for this somewhat unpleasant result is that we have de­ signed the transform to minimize costs for steady-state operation. This will not, in general, minimize cost when the system is passing from one steady-state to another. Clearly, for 6L(t) = TA , a constant, (t ^ 0), we want

(4.20) ju(0 = T + a transient term,

(4.21) 80(t) = 0 + a transient term.

The transient term in (4.20) should be such that /*(<) will not over­ shoot that is, the system should be over-damped. This implies that the roots of the characteristic equation of Z(p) should be negative and real.

To get the desired steady-state behavior of /*(*) for the indicated load we require that Lim p_»o pZ(p)(T/p) — Lun p_o Z(p)T = T. From (4.21) we infer that 00(0 should be heavily damped, with Lim p_o Y(p)T = 0. From (4.14) we see that the latter condition is a sufficient condition that Lim p_0 Z(p)T = T.

As can be seen from inspection, these limiting conditions are satis­ fied by the transforms of (4.18) and (4.19) although, because of in- stability, the conditions on the transients are not. To remedy this situation we replace the denominator of Y(p) by (\/pP +\/a)2. The resulting transform,

is critically damped and approaches the transform of (4.19) for large p. The characteristic equation has the two equal negative real roots: Po = (O/P)*. The transient term in p(f) will be of the form Ate~Pot.

Construction of the decision rule. We return now to the problem of finding a K3 and K2 that will realize the Y(p) of (4.22). We will first explore the simple case where KI = 1 (no production lag). In this case

(4.23)

If we now set K* = 1 pp, Kz = pp2 + [2(p<r)* — l]p + a and substitute in (4.23), the result is (4.22). Moreover, we will have for

(4.24) Z(p)

n8ERVOMECHANISM THEORY IN PRODUCTION CONTROL 265

Since, in the case of zero tune lag, M(*) = n(0> (4.24) gives the follow­

ing rule for determining ij(0 :

(425) P

In the case where there is a fixed production lag, K4 = e"1*, we have

u^(4.26)

we define X(p) = i?(p)/»t(p), we obtain from (3.2) and (4.26)

Giving K2 and K* the same values as hi the previous case, we get

/AOQN y/ N (4.28) X(p) - e *)

The corresponduig decision rule is

- r)

This rule we may take as a realizable approximation to the rule that

would minimize costs. For the limiting cases as p » » and p > 0, it

has the same properties as the rule derived from (4.24).

5. FURTHER CONSIDERATION OP THE COST CRITERION

The cost criteria developed in Section 4 are undoubtedly greatly

oversimplified. In this section we will consider possible methods of

constructing a more realistic criterion. In particular, we wish to intro­

duce a more complete analysis of that part of the cost function that de­

pends on rate of manufacture.We suppose that the cost of manufacture is the sum of three com­

ponents:1. Variable costs proportional to the rate of manufacture (e.g., cost

of materials). Since these costs are determined by the number of orders

to be filled, and hence are independent of the control system, we may

continue to ignore them.2. Fixed costs proportional to plant capacity, i.e., to the maximum

rate of manufacturing activity. The previous section indicates how

these costs can be handled in designing the control system.

3. Sticky costs proportional to the rate of manufacture when this

is constant, but not capable of being reduced immediately as the rate

266 H. A. SIMON

of manufacture declines. In first approximation we may assume that as the rate of manufacture increases from a stable level, sticky costs will increase proportionately, but that if the rate of manufacture decreases, there is a fixed upper limit to the rate at which sticky costs will de­ crease.

Suppose (Figure 4) that M(<) is subject to oscillation of period 2/a and amplitude A. The slope of M will be =feAa. Suppose, further, that sticky costs, £(0 can only decrease with a slope of j8, but can increase as rapidly as p.

The ordinate of /t(0) is A; the slope of /*(<) is Act in the interval 0 ^ t < I/a, Aa in the interval I/a ^ t < 2/a. The ordinate of {(0) is A-, the slope of £(0 is 0 in the interval 0 ^ t < ta ; Aa in the in­ terval ta ^ t < 2/a. The integral of £(0 - /i(<) over the interval 0 ^ t < 2/a is then the area of the triangle whose vertices are indicated

t=0

FlGUBE 4

by dots. This area is £ the ordinate £(!/<*) times ta : £(l/a) = A — (0/a). The value of ta is given by

(5.1) Aa[ta - (I/a)] = A -

or

AaHence,

(5.3)*'* Aa- 2A A (Aa - |8)

Aa + ft a (Aa + ft)'

If 1 is an integral multiple of 2/a, then

J (£ - M) dt 2 2 (Aa

But, for this same interval, / ndt = A/2-, hence we have for the

ratio of sticky costs to production

(5.5)2 (Aa - 0) A " (Aa + 0)

O'

SERVOMECHANISM IN PRODUCTION CONTROL 267

It follows that sticky costs will be increased by an increase in the amplitude of the oscillations of n(t) behaving, in this respect, like fixed costs and will also be increased by an increase in the frequency of it. In designing our optimal criterion we disregarded this latter con­ sideration. Hence, the design of the decision rule can be unproved by decreasing the response of n(t) to 6L(f) for high frequencies of the latter at the expense of increasing somewhat the response of Qo(f). Again it is reassuring that our results coincide with common sense.

Assumption of a sinusoidal oscillation of p and of £ leads to the same kind of result. Finally, fixed costs and variable costs can be subsumed as a limiting case of sticky costs by considering a range of different cost categories, each with its characteristic 0. The /3 for variable costs would be infinite; for fixed costs, zero. If we can define some kind of an average /3, this can be used as a basis for our criterion of manufacturing costs.

6. CONCLUSION

The general conclusion to be drawn from our explorations, however -. tentative these have been, is that the basic approach and fundamental

f techniques of servomechanism theory can indeed be applied fruitfully to the analysis and design of decisional procedures for controlling the rate of manufacturing activity. To be sure, most of the conclusions wet^have reached could, at least in a qualitative sense, be reached intui­ tively. But even here, intuition has been aided by the frame of reference

^ that servomechanism theory provides. Moreover, the more exact pro­ cedures permit statement of our results with a degree of precision that could not be attained without them. Even in this very early stage the theory permits actual numbers to be inserted for the construction of specific decision rules that would apply, with a considerable degree of realism, to actual situations.

Carnegie Institute of Technology7. REFERENCES

The following list of references covers some of the more systematic and lucid introductions to servomechanism theory. They are listed in order from the relatively elementary treatments to the more advanced or specialized.

[1] LATTER, H., R. LESNICK, AND L. E. MATSON, Servomechanism Fundamentals, New York: McGraw-Hill, 1947. An introductory treatment employing dif­ ferential equations rather than the Laplace transform. Analyzes in detail the behavior of a few very simple engineering servos, and gives a clear

< picture of the servo concept.[2] JAMES, H. M., N. B. NICHOLS, AND R. S. PHILLIPS (eds.), Theory of Servo-

mechanisms, New York: McGraw-Hill, 1947. Chapter I, "Servo Systems," gives a good introduction to basic concepts. Chapter 2, "Mathematical Background," gives an excellent introduction to the Laplace transform,

268 H. A. SIMON

its physical meaning, and its relation to the weighting function. General

design principles and techniques are discussed in Chapter 4, and more ad-

! vanced topics in other chapters.

\ [3] GARDNER, M. F-v AND J. L. BARNES, Transients in Linear Systems, Vol. I,

; New York: John Wiley and Sons, 1942. A clear, systematic exposition of

: Laplace transform theory and methods.

i [4] BROWN, G. S., AND D. P. CAMPBELL, Principles of Servomechanisms, New

j York: John Wiley and Sons, 1948. Parallel to [3], but with more emphasis

I on system design, and less on analysis.

; 15] McCoLL, L. A., Fundamental Theory of Servomechanisms, New York: Van

Nostrand, 1945. An elegant brief treatment stressing fundamental concepts

and employing Laplace transform methods.

\ [6] WIENER, N., The Extrapolation, Interpolation, and Smoothing of Stationary

I Time Series, New York: John Wiley and Sons, 1949. An approach to the

problem of forecasting a stochastic input or load. (See also [2], Chapters

\ VI-VIII.)

i (7] OLDENBOURG, R. C., AND H. SARTORIUS, The Dynamics of Automatic Con-

\ trols, American Society of Mechanical Engineers, 1948. A systematic

1 treatment of controllers, using Laplace transform methods. Includes ex-

; tensive discussion of fixed lags, nonlinearities, and discontinuous regula-

] tion. (On the last point, see also [2], Chapter V; [5], Chapter X and Ap-

; pendix; and [8], Chapter IX.) X'N

{ Excellent bibliographies will be found in [3], [4], [5], and [7]. For dis-

] cussions and examples of the use of Servomechanisms in the study of

| economic systems see all the following:

I [8] COOPER, W. W., "A Proposal for Extending the Theory of the Firm," Quar­

terly Journal of Economics, Vol. 66, February, 1951, pp. 87-109.

18] MOREHOUSE, N. F., R. H. STROTZ, AND S. J. HORWITZ, "An Electro-Analog

Method for Investigating Problems in Econometric Dynamics: Inventory

Oscillations," ECONOMETRICA, Vol. 18, October, 1950, pp. 313-328.

[10] ENKE, STEPHEN, "Equilibrium Among Spatially Separated Markets: Solu­

tion by Electric Analogue," ECONOMETRICA, Vol. 19, January, 1951, pp.

40-47.[11] WIENER, NORBERT, Cybernetics, New York: John Wiley and Sons, 1948.

[12] GOODWIN, RICHARD M., "Econometrics in Business-Cycle Analysis," Chap­

ter 22 in Alvin H. Hansen, Business Cycles and National Income, New

York: W. W. Norton and Co., 1951.

1