Sales-advertising models, population dynamics, optimal control, distributed parameter systems, diffusion models. I. Introduction. The problem of optimizing ...
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 52, No, 3, MARCH 1987
Dynamics and Optimization of a Distributed Sales-Advertising ModeP T. I.
SEIDMAN, 2
S. P.
SETHI, 3 A N D
N. A.
DERZKO 4
Communicated by R. Rishel
Abstract. A general sates-advertising model is developed in which the state of the system represents a population distribution over a parameter space. With appropriate interpretations, this can include income, family size, geographic distributions, etc. Effects of information diffusion, interaction, and population migration are included. Under fairly general conditions, it is shown that such model are well posed and that there exists an optimal control. Key Words. Sales-advertising models, population dynamics, optimal control, distributed parameter systems, diffusion models.
I. Introduction The problem of optimizing advertising expenditures of a firm over time is one of central importance in the field o f marketing. Since the profits arise from sales, a formulation of such an optimization problem requires consideration of the dynamical relationship between sales and advertising, in other words, a sales-advertising response model, Many different sales-advertising models have been proposed in the literature. These models have been surveyed in Little (Ref. 1) and Sethi (Ref. 2). One class of response models can be termed diffusion models (Refs. 3-6). The fundamental concept, here, is that of the market, but there will be some period of time during which individuals learn of the advertisement This work was supported by NSERC Grant No. A-4619and by Grant No. AFOSR-82-0271. Thanks are due to G. Haines and T. Mitchell. 2 Professor, Department of Mathematics and Computer Sciences, University of Maryland Baltimore County, Catonsville, Maryland. 3 Professor, Faculty of Management, University of Toronto, Toronto, Canada. 4 Associate Professor, Department of Mathematics, University of Toronto, Toronto, Canada, 443 0022-3239/87/0300-0443505.00/0 C/:}1987 Plenum Publishing Corporation
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or, more precisely, become favorably disposed toward the firm's brand by coming into contact with an advertisement or by word of mouth. Such a process leads to the definition of goodwill or the cumulative effect of advertising at each moment as the number of individuals who are favorably disposed toward the firm's brand. One such diffusion model was proposed by Vidale and Wolfe (Ref. 3); see also Gould (Ref. 5) and Blattberg and Jeuland (Ref. 6). Vidale and Wolfe proposed that the firm's advertisements act on those peopole who are unaware of or are not favorably disposed toward the firm's brand and convert a fraction of this into favorably disposed individuals. Another diffusion mechanism was proposed by Ozga (Ref. 4). Here, the information spreads by word of mouth interaction, rather than by an impersonal advertising medium. That is, individuals who are favorably disposed toward the firm's brand convince other individuals to-become favorably disposed at a certain rate, and vice versa. In addition to these diffusion effects, there is also a decay of favorably inclined individuals, either because of forgetting or because of exogenous effects, such as advertising by other firms. Optimization problems with these diffusion advertising models were treated by Gould (Ref. 5) and Sethi (Refs. 7 and 8). In this paper, we develop a distributed diffusion model of sales-advertising response that is sufficiently general to include the models that have been cited so far. Here, population is distributed over several parameters such as income levels, geographic location, family sizes, etc., and we now consider migration of population over the parameter space as well as the diffusion mechanisms mentioned above. We introduce our notation and formulate the model in the next section. In Section 3, we prove that the model is well posed under fairly general conditions. In Section 4, we introduce an optimization problem for the distributed-parameter control problem and show that there exists an optimal solution to this problem under fairly general conditions. These existence questions would be routine for a finite-dimensional model, but the lack of local compactness for an infinite-dimensional state space introduces a novel technical difficulty. It should be emphasized that we have attempted here to construct an extremely general class of models. Hence, any particular application of this must involve a much more detailed specification of an individual model within this class. The present work is to be viewed not as presenting a directly usable formulation, but rather as developing a general setting within which such formulations can be treated. In that sense, this is the introduction to a series of papers condsidering such additional aspects as: (i) the extent to which distributed models can be approximated by compartmental, multi-
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sector models; (ii) methods for analytic or numerical computation of nearoptimal control strategies; and (iii) inverse problems of determining appropriate parameters of specific models for particular applications.
2. Notation and Model
We develop the dynamics of population and that of goodwill. The treatment o f the latter follows the spirit of the Vidale-Wolfe advertising model (Ref. 3), but now in the context of a distributed population and including the effects of population transfer and word of mouth advertising (Refs. 4 and 5). It is convenient to develop the population model first and then the advertising model. We shall need several definitions and first describe these without precise assumptions. These assumptions will be imposed as needed in Sections 3 and 4.
Population Model.
We introduce the following definitions:
f~ =general parameter space, taken with a natural measure dx. We assume f~ to have finite measure. We note that the specification of x ~ f~ includes such population variables as may be relevant: geographic, socioeconomic variables, etc. B = Banach space L~(f~). This class of essentially bounded functions is sufficiently large to include practical applications and, at the same time, allows us to prove the existence of an optimal advertising policy for our model. B+ = { f c B: f_> 0}.
P(t, • ) c B+ denotes population density on ~] at time t 6 R+, so that the population in X C Y / a t time t is ~x P(t, x) dx. a(x, t, P(t,. ) ) = rate of immigration to x ~ ~ from outside the system. fl(x, y, t, P(t, • ) = birth rate; i.e., births at x due to population at y. O(x, y, t, P(t,. )) = rate of migration from y to x. y(x, t, P(t,. ) ) = combined death and emigration rate out of x c ~q.
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It is convenient to introduce the combinations
q(x,y, t, P(t, .)) =fl(x,y, t, P(t, .))+ O(x, y, t, P(t, .)),
fl)
c(x, t, P(t," )) = f O(y, x, t, P(t," )) dy+ 3,(x, t, P(t," )). Jlz
(2)
With these definitions, we can write the following evolution equation for the population density:
aP(t,x)/Ot=J
f
q(x,y, t, P(t, .))P(t,y) dy
-c(x, t, P(t, . ))P(t,x)+ce(x, t, P(t, .)).
(3)
Several remarks are in order at this point. The principal modeling feature is the assumption that population transfer (migration) is essentially instantaneous. That is, each transfer is viewed, perhaps more realistically, as a single step with no trajectory or consideration of intervening points. This nonlocal property of the migration process removes the need for introducing a topology on f~ at this stage of the formulation. This feature, however, comes at some cost; e.g., it precludes the possibility of simply taking age to be a component of x e Ft, although this would involve only minor modification. In the interest of simplicity, we have also chosen to ignore the dependence on the past history of P for the various rates defined above. Note, however, that the paper could be extended to include these additional factors if desired. We would also like to point out an interesting alternative interpretation of the same mathematical model. In this version, P(t, x) would refer to density of families, so that the parameter x indicates such variables as family size, its age distribution, income, etc. Here, the birth of an individual is given by a migration in the parameter space. Likewise, an individual moving out would give birth to a new family, together with a migration of original family in this parameter space.
Advertising Model. We now proceed to develop the advertising model. For this, let A(t, x)c B÷, with O0, meaning both A >- 0 and P - A -> 0, the terms preceded by + signs in (14) are also ->0. Furthermore, on any fixed interval [0, T], we can use (10) to find a constant M such that M - d - {~_H(P - A)} --.>0,
M-C-{
V + E+FIA} >_O.
Thus, for such M, we have
y > - 0 0 M y + f (t, y) >_O,
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where f(t, (A, P-A)+), + meaning transpose, is the right side of (14). It follows by Corollary 3.1 that
r
l
Y(t)=LP(t)_A(t)j>-O,
f o r 0 -0 and dy/dt =f(t, y) on [0, co), that f satisfies the hypothesis of Theorem 3.2, and that y -> 0 implies f(t,y)>-O on the domain o f f . Then, y(t)>_O for t - - 0 . Proof. Note that the Picard mapping (17) takes the closed set B+ of nonnegative functions into itself. Recall from the proof of Picard's theorem that the solution of (16) is locally the limit of iterates of T applied to a starting function with constant value y(0). It follows from our hypothesis that all the iterates belong to B+. Consequently, y(t) is also in B+. []
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The following corollary shows that the hypothesis on f can be weakened. Corollary 3.1. L e m m a 3.1 is still true if the hypothesis y >- 0 ~ f ( t , y) -> 0 is weakened to read: there exists a constant M such that y - 0 implies
My+f(t,y)>-O. Proof.
Observe that z = exp(Mt)y satisfies
z'= exp(Mt)(M e x p ( - M t ) z + f(t, exp(-Mt)z)) ~=h(t, z), where
z >-O ~ e x p ( - M t ) z >_0 0 h (t, z) >-O, by hypothesis. L e m m a 3.4 now applied to yield z(t) >-O, from which y(t) >-0 follows. []
4. Existence of Optimal Controls In this section, we consider Eq. (8) for A, the favorably disposed individuals, and treat all its data except v as fixed. The advertising effort v is allowed to be a controller. We let A(v, t, x) denote the solution of (8) arising from v. Since we do not refer explicitly to the x dependence, the notation is shortened to A(v, t). Our purpose is to prove, under reasonably general assumptions, that there exists a controller v(t), 0 -< t < oc, which maximizes the profit functional
J(v) =J(A(v," ), v)=
f0 o exp(-rt)[~r(t,
A(v, t ) ) - g ( t , v)] dt.
(18)
Here, r respresents the constant discount rate. The function ~r(t, A) represents the rate of gross profit margin less fix costs at time t when the stock of goodwill is at level A. The variable cost function g(t, v) is the cost of advertising at time t when the rate of advertising is at level v. Thus, the functional J(v) represents the present value of net profit streams over the infinite horizon. The precise assumptions and notation follow shortly. Since the detailed structure of the terms in Eq. (8) is not relevant to our analysis, we rewrite it as
OA/Ot= F(t, A, v) = Fo(t, A)+v(t)Fl(t, A), where
F1(t, A) = P ( t ) - A ( t ) and Fo(t, A) contains the rest of the terms on the right side.
(19)
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We retain the assumptions (9) of Section 3. Under these, it is readily verified that F~(t,A), i = 0 , 1, is continuous in t and Lipshictz in A. The following further assumptions and definitions are needed.
Assumptions and Definitions. We introduce the following assumptions and definitions. R~ is the range space for the controllers v(t). Rv C B is assumed to be compact in the topology of B and convex. Compactness is essential for our existence proof and sufficiently general to include the case when advertising effort is allocated among a finite number of modes. In this case, Rv is finite dimensional and therefore compact. U is the space of admissible controllers on 0 < t < oo, that is, the space of measurable R~-valued functions. The phrase "v admissible" is used interchangeabty with v e U. U, is the space of admissible controllers on [0, z].
(20)
r satisfies r > Mo,
(21)
where MQ is the exponent appearing in (10); that is, the discount rate must exceed the rate of population increase. ~r(t, A) is a continuous function of t and satisfies a global Lipschitz condition as a function of A on [0, co) x B. (22)
g( t, v) satisfies o'~ e x p ( - r t ) sup g(t, v) dt 0. Since vj takes values in a precompact set, {vj} is weakly sequentially compact in the space LP([0, rl], B). Thus, there exists subsequence {vile}, tending weakly to a limit t5 on [0, ~]. By choosing Tk-->OOmonotonically, we find successive subsequences {v~k)}, each tending weakly to a limit function 75 on [0, rk] in the space L p. The diagonal sequence v9 = vy ~ then defines a limit function 15 on [0, co) with the property that t) -->~5weakly in L p for each ~rk. On any fixed interval [0, r], the corresponding {Aj} constitute, by Lemma 4.1, a uniformly equicontinuous family of functions taking values in the precompact set ~¢, which, by the Ascoli-Arzela theorem (Ref. 12, pp. 266 and 392), must itself be precompact. Once more, by a successive subsequence process, we extract a diagonal subsequence Aj(t), which tends, in the lit" II1-sense, to a limit function A(t) on each [0, %]. It is clear that A(t) is measurable and satisfies 0-< A(t) -< P(t). For the remainder of the proof, we rename the sequence of pairs (vs(;), As(;)) to be simply (vj~ Aj). It is immediate that ] & lim J(Aj, vj) = sup{J(v): v c U}. j ,.~ oo
It remains to prove that (i)
~ is admissible;
(ii)
A(t)=A(TS, t);
(iii)
] = J(A, ~5).
Assertion (i) follows, because R~ has been assumed closed (in the strong topology and hence, by convexity, in the weak topology as well), so that, t a.e., one has ~(t) c R~. To prove assertion (ii), we fix r > 0 and note that
As(t) = Ao+
F(s, Aj, vs(s)) ds.
(29a)
We multiply (29a) by a bounded measurable test function ~b of compact support and integrate to obtain
=
f/It;
]
F,(s, Aj) ds vj(t)¢(t) dt,
(29b)
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Since Aj -->4 and F0, F~ are uniformly Lipschitz for 0--- s -< r, it follows that F~(Aj, .)--> F~(4, .) in C~, i--0, 1. Also,
dp(t) F~(s,A~)-->~(t)
fo
f/ F,(s, A) as,
in C. From these observations, we conclude that the right side of (29b), being the inner product o f a strongly and a weakly convergent sequence, converges to
while the left side converges to
ff q~(t)[4(t)-ao-f2 Fo(4, s) ds]dt,
(31)
so that (30) and (31) are equal for a dense set of test functions ~b. It follows that
4(t) = Ao+
;o
[Fo(4, s) + F1(4, s)e(s)] at,
a.e.,
(32)
so that 4 = A ( g - ). We proceed with assertion (iii). By definition,
J(Aj, vj) =
I)
=
exp(-rt)[~r(t, Aj) - g(t, vj)]
exp(-rt)~r(t, A i) dt-
dt
fo o exp(-rt)g(t, vj) dr.
(33)
0
We fix r > 0 and deal with the first term. Since Ai ~ 4 in C~ and
IfoeXp(-rt)[Tr(t, Aj)-~r(t, 4)]dtl
fooexp(-rt)l[A~(t)-,4(t)ll dt
