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Optimal Control of the VidaleWolfe AdvertisingModel
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Operations Research · May 1972
DOI: 10.1287/opre.21.4.998
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Suresh SethiUniversity of Texas at Dallas
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OPTIMAL DYNAMICS OF THE VIDALEWOLFE ADVERTISING MODEL
PART I: FIXED TERMINAL MARKET SHARE
by
Suresh P. SethiTECHNICAL REPORT 729May 1972
DEPARTMENT OF OPERATIONS RESEARCH
Stanford University
Stanford, California
Research and reproduction of this report was partially supported byOffice of Naval Research N0001467A01120011; National ScienceFoundation Grant GP 31393; and U.S. Atomic Energy Commission Contract
AT(043)326PA #18.
Reproduction in whole or in part is permitted for any purposes of
the United States Government. This document has been approved for
public release and sale; its distribution is unlimited.
OPTIMAL DYNAMICS OF THE VIDALEWOLFE ADVERTISING MODEL
PART I: FIXED TERMINAL MARKET SHARE
by
Suresh P. SethiStanford University
I.
Introduction.
As one of the earliest management science applications to
marketing, Vidale and Wolfe [1.4] developed a simple model of the
sales response to advertising that was consistent with their experi
mental observations. In the fifteen years since its inception, the
model has had a wide appeal because of its extreme simplicity andcapacity to capture the interaction between advertising and sales in
an intuitively satisfying manner.
Vidale and Wolfe argued that changes in the rate of sales
of a product depend on two effects; response to advertising whichacts (via the response constant p) on the unsold portion of themarket, and loss due to forgetting which acts (via the decay constant k)
on the sold portion of the market:
(1.1)
x = pu(lx)  kx;x(0) = x0
I would like to thank Professor J. V. Breakwell, R. E. Turner,
and C. P. Neuman for their helpful suggestions. An earlier version
of this paper with Part II was presented at the 41st ORSA Conference
at New Orleans, April 1972.
1
where x is the market share (i.e0,,the rate of sales expressed as a
1
fraction cf the market potential or saturation
level) and u is
the rate of advertising expenditure (a
control
variable), at time t.
Whereas Vidale and Wolfe offered their model primarily asa description of actual market phenomenar_epresented by cases whichthey had observed [14], the present paper (Part I) obtains the optimaladvertising schedule (optimal control) of the model; in the sense
that it must maximize a certain objective function over horizon T
while attaining a specified terminal market share
XT
at time T.2
In the next section, we complete the statement of the problem bydefining the objective function.
2.
Optimal Control Problem
,
It is assumed that the objective to be maximized is the
present value of the profit stream up to the horizon T. Morespecifically, if n denotes the maximum sales revenue potential
(this assumes a constant margin per unit product) and i, the discount
rate, then the optimal control problem is:
T(2.1)
max
J =
[nx(t)  u(t)] e
it
dt
u(t) = 0
0
1
This represents a slight departure from the VidaleWolfe model [14].
In terms of their formulation x(t) = S(t)/M, where S(t) is therate of actual sale at time t and M is the saturation level.
2
See Section 7(i) for the problems treated in Part II [12].
2
subject to (1.1) and the terminal constraint x(T) = x
T
o
Note that the
requirement 0 < x(t) < 1 for all t is automatically satisfied if
0 < x
0
< 1, x
0
being the initial market share.
A convenient procedure to solve this problem, where there
is only one state variable and where the control u appears linearlyin both the state equation (L1) and the objective function (2.1),is by using Green's theorem
181.
This is carried out in the following
section.
3.
Solution by Green's Theorem
.
First we substitute
(3.1)
udt _ dx + kx dt
p (lx)
derived from (1.1) into the objective function (2.1) to obtain the
line integral
it
(3.2)
J =
[
7x
_.
kx
]
e
t dt dx
p(1x)
p(1x)
Using Green's theorem, we can express the line integral above as a
double integral in (t,x) space;
it
.
(3.3)
J
= f
[

e
]

a
[Trx 
kx
]
eit
dt dx
a
t
p (1x)
ax
p (
ix)
=
1
a
i
f
[
k
+
1

Trp ] eit dt dx
p
(1x)2
1x
9