=Paper=
{{Paper
|id=Vol-2018/paper-12
|storemode=property
|title=The Dynamic Model of Advertising Costs with Continuously Distributed Lags
|pdfUrl=https://ceur-ws.org/Vol-2018/paper-12.pdf
|volume=Vol-2018
|authors=Igor Lutoshkin,Nailya Yamaltdinova
}}
==The Dynamic Model of Advertising Costs with Continuously Distributed Lags==
https://ceur-ws.org/Vol-2018/paper-12.pdf
The Dynamic Model of Advertising Costs with
Continuously Distributed Lags?
Igor Lutoshkin and Nailya Yamaltdinova
Ulyanovsk State University, Ulyanovsk, Russia,
www.ulsu.ru
Abstract. The dynamic optimal control problem of advertising costs
in case of a company’s limited advertising budget is analyzed. Initial
optimization problem is formulated as a system of nonlinear integral
equations of Volterra type. The constructed model takes into account
the accumulated advertising effect and the accumulated effect of previ-
ous sales on the consumer demand. The accumulation of these effects is
supposed to be distributed on an interval of time from the beginning of
the planning period to a given point in time, that is, the delay of con-
sumer reaction on advertising does not take the specified value and tends
to infinity in the long run. Also the equation of consumer demand is de-
termined. The total profit maximization problem under restrictions is
stated. The existence theorems for the solutions of the demand equation
and the total profit maximization problem are proved. The mathemat-
ical model can be integrated in information systems of enterprises for
obtaining an optimal decision of a problem of the best distribution of
advertising costs. In addition, the mathematical model is tested with the
real data. In this case the Pontryagin’s maximum principle and numerical
methods of solving the integral and differential equations are used.
Keywords: computer modeling, mathematical model of advertising, dis-
tributed lag, optimal control, Pontryagin’s maximum principle, existence
of solution
1 Introduction
The main purpose of any commercial company is to satisfy clients, encourage
them to repeat purchases and to recommend company’s products to other po-
tential clients. In this case, it is important to increase brand’s recognition in
order to attract new customers. Also a company should keep the interest of old
clients. One of the most effective and popular tools which can help to achieve
those goals is advertising. Generally, the advertising costs should be included in
the total costs as well as other company’s costs. But advertising does not require
any change in production technologies or quality of goods, and it makes sense
to consider advertising separately from other costs.
?
This work has been done as a part of the state task of Ministry of Education and
Science of the Russian Federation No 2.1816.2017/PP.
Advertisement can influence on client’s decisions immediately, but most of
them have a ’lagging’ nature. In addition, there are other factors which have an
impact on sales [10,12,14], e.g. quality of goods, company’s reputation etc. These
factors make clients repeat purchases and create the effect of previous sales.
Accordingly, the demand is changing under the impact of the accumulated
effect of advertising costs and the accumulated effect of previous sales.
Obviously, an advertising budget is usually limited to some amount of money.
Thus, the problem of the best distribution of advertising costs over the planning
period can be considered as a dynamic optimal control problem. In this case,
the total profit function can be chosen as an objective function which needs to
be maximized.
The optimization advertising models were considered in [1, 3, 4, 6, 8, 11, 13].
Many modern advertising models are based on them. They are discussed by Jian
Huang, Mingming Leng, Liping Liang in detail [5].
In our paper, in addition to advertising costs, other factors of influence on
consumer demand are taken into account. The delayed consumer reaction is
considered here, too. The paper also includes practical application. It is supposed
that the consumer demand is non-linear with respect to the accumulated effect
of advertising costs and the accumulated effect of previous sales. The practical
application of the dynamic model includes the algorithm for solving the problem
of the best distribution of advertising costs by using the Pontryagin’s maximum
principle [2] and some numerical methods for solving integral and differential
equations.
2 State of the Problem
Let y(t) is the revenue function at the moment t, u(t) is the advertising costs,
v(t) is the function which determines the accumulated advertising effect, w(t)
determines accumulated effect of previous sales:
Z t+λ
v(t) = Gu (τ )u(t − τ ) dτ ; (1)
0
Z t+λ
w(t) = Gy (τ )y(t − τ ) dτ. (2)
0
Here λ is the length of the time interval of influence of previous sales and
previous advertising costs on y(t). The revenue at the moment t depends on v(t)
and w(t):
y(t) = f (v(t), w(t)). (3)
The functions Gu (τ ), Gy (τ ) describe the effects of previous advertising costs
and previous sales [10]. There are some assumptions about the functions and
f (v, w) :
1. If v and w take small values (a market isn’t saturated by company’s products
and advertising is positively received by consumers), the function f (v, w)
increases. But consumers’ reaction may change after some time [1, 9], and
the function f (v, w) may become non-increasing with respect to v. Also,
f (v, w) may decrease because of market saturation and supply constraints
of the company. Thus, the function f (v, w) is concave with respect to w.
2. Until the certain moment τu∗ the advertising influence on the demand in-
creases. After that, the advertising effect begins to decrease until it disap-
pears. The possibility of negative advertising effect is excluded, i.e. advertis-
ing costs of the company do not reduce product demand. Thus, the function
Gu (τ ) is non-negative and it has unique maximum τ = τu∗ which is the mo-
ment of the highest advertising effect. If this function is differentiable, then
assumption is equivalent to the following conditions:
Gu (τ ) ≥ 0, ∀τ ∈ [0; +∞); lim Gu (τ ) = 0;
τ →+∞
G0u (τ ) ≥ 0, τ ∈ [0; τu∗ ); G0u (τ ) ≤ 0, τ ∈ (τu∗ ; +∞).
3. Also we can note that consumers repeat purchases because of their posi-
tive experience. In this case, the consumer demand becomes higher and the
function Gy (τ ) increases. However, the experience of the first purchases is
usually forgotten. It affects the current purchase weakly, giving place to the
recent experience. Therefore, the function Gy (τ ) has the properties:
Gy (τ ) ≥ 0, ∀τ ∈ [0; +∞); lim Gy (τ ) = 0;
τ →+∞
G0y (τ ) ≥ 0, τ ∈ [0; τy∗ ); G0y (τ ) ≤ 0, τ ∈ (τy∗ ; +∞).
The financial result of the company is the profit or loss which is determined
by the condition π(y(t), u(t)) = y(t) − c(y(t), t). Here c(y(t), t) is the function
of total costs which include fixed and variable costs. Thus, we can write the
objective function:
Z T Z T
Π(T ) = π(y(t), u(t)) dt = (f (v(t), w(t)) − u(t) − c(y(t), t)) dt
0 0
where c(y(t), t) = c1 (y(t), t) + c2 , c denotes total costs of the company which
are connected with the production of goods excepting the advertising costs, c1 -
variable costs, c2 - fixed costs.
It is logical that variable costs are in direct ratio to the volume of output
with the rate µ. We have following:
Z T Z T
Π(T ) = π(x(t), u(t)) dt = ((1 − µ)f (v(t), w(t)) − u(t)) dt. (4)
0 0
Let the advertising budget be limited in the following way:
0 ≤ u(t) ≤ b, t ∈ [0; T ]. (5)
The main task of the company is to maximize profit taking into account the
fixed restrictions. So, we can formulate the following optimization problem: to
maximize the functional (4) under the conditions (1), (2), (3), (5).
Rewrite (1) and (2):
Z t Z t
v(t) = Gu (t − s)u(s) ds, w(t) = Gy (t − s)y(s) ds,
−λ −λ
where s = t − τ.
Denote by φu (t), φy (t) the following functions:
Z 0 Z 0
φu (t) = Gu (t − s)u(s) ds, φy (t) = Gy (t − s)y(s) ds.
−λ −λ
The functions v(t), w(t), Π(t) can be represented:
Z t
v(t) = φu (t) + Gu (t − s)u(s) ds, (6)
0
Z t
w(t) = φy (t) + Gy (t − s)f (v(s), w(s)) ds, (7)
0
Z t
Π(t) = ((1 − µ)f (v(s), w(s)) − u(s)) ds. (8)
0
Thus, the maximization of Π(T ) under conditions (5), (6), (7), (8) is equiva-
lent to the maximization of (4) under conditions (5), (3), (1), (2), and it presents
the optimal control problem with integral equations of Volterra type.
3 Existence of the Solution
Suppose that advertising costs function u(t) is piecewise-continuous on the right
in [0, T ].
Consider the question of the solution existence of the equations (6), (7), (8).
If Gu (τ ) is continuous, then v(t) is continuous in [0; T ]. Considering (5), (6)
we can state the existence of the value b1 > 0 :
Z t Z t
0 ≤ φu (t) + Gu (t − s)u(s) ds ≤ max φu (t) + Gu (t − s)b ds ≤ b1 .
0 0≤t≤T 0
Thus, the function of the accumulated advertising effect satisfies the con-
dition v(t) : 0 ≤ v(t) ≤ b1 for any advertising strategy u(t) that satisfies the
condition (5). The problem of the solution existence of the equation (7) is solved
by the theorem 1 [10].
Theorem 1. Let the functions Gu (τ ) ∈ C([0; T ]), Gy (τ ) ∈ C([0; T ]), the func-
tion f (v, w) is continuous and it satisfies a Lipschitz condition with respect to
w for all w. Then, for any piecewise-continuous function u(t), t ∈ [0; T ] that
satisfies the restriction (5), there exists continuous and unique function w(t),
t ∈ [0; T ] that satisfies the condition (7).
Proof. The proof of the solution existence of the linear equation of Volterra type
is described in [7]. Based on this statement we prove a solution existence of the
integral equation (7).
Initially, the function f (v, w) satisfies the Lipschitz condition with respect
to w. That is, there exists the constant L : |f (v, w1 ) − f (v, w2 )| ≤ L|w1 − w2 |,
∀w1 , w2 . Define the operator A in the following form:
Z t
Aw(t) ≡ φy (t) + Gy (t − s)f (v(s), w(s)) ds.
0
Evidently, there is a finite number M = max Gy (τ ). Due to properties of
0≤τ ≤t
the function Gy (τ ), there is max Gy (τ ) which is equal M. We can write the
0≤τ ≤t
following inequality:
|Aw1 (t) − Aw2 (t)| =
Z t
= Gy (t − s)(f (v(s), w1 (s)) − f (v(s), w2 (s))) ds ≤
0
≤ M Lt max |w1 (s) − w2 (s)|.
0≤s≤T
Introduce Ak , k is the multiple successive applying of the operator A. In this
case A2 w ≡ A(Aw), Ak w ≡ A(Ak−1 w). We have the inequality:
A2 w1 (t) − A2 w2 (t) =
Z t
= Gy (t − s)(f (v(s), Aw1 (s)) − f (v(s), Aw2 (s))) ds ≤
0
Z t
(M Lt)2
≤ ML |Aw1 (s) − Aw2 (s)| ds ≤ max |w1 (s) − w2 (s)|.
0 2 0≤s≤T
Similarly,
(M Lt)k
Ak w1 (t) − Ak w2 (t) = max |w1 (s) − w2 (s)|.
k! 0≤s≤T
Using the metric in the space of continuous functions
ρ(w1 , w2 ) = max |w1 (s) − w2 (s)|
0≤s≤T
it is possible to write:
(M Lt)k
ρ Ak w1 , Ak w2 ≤ ρ(w1 , w2 ).
k!
(M Lt)k
Evidently, there exists k : < 1. It means, that operator Ak is contract-
k!
ing. Thus, the solution w(t) of the equation (7) exists, it is unique and continuous
in [0; T ]. t
u
Theorem 1 gives the conditions of the existence of the global solution of the
equation (7).
Remark 1. Let us suppose that the f (v, w) is non-negative, concave and non-
decreasing monotonously with respect to w. If the finite partial derivative fw0
exists with respect to w = 0, and the function f (v, w) satisfies the Lipschitz
condition with respect to w for any w, then the solution of the equation (7)
exists and it is non-negative in [0; T ].
Consider the optimization problem: to maximize Π(T ) under the conditions
(5), (6), (7), (8).
A finite value of the accumulated profit Π(T ) is a functional of control u(·).
Introduce J(u(·)) ≡ Π(T ). Formulate the theorem of the solution existence of
the optimization dynamic problem.
Theorem 2. Assume that the conditions of the theorem 1 are fulfilled, f (v, w)
does not decrease monotonously with respect to v; then there are two alternatives:
1. There exists {u∗ (t), 0 ≤ t ≤ T } : (5), a solution to the equations (6), (7),
(8) {v ∗ (t), w∗ (t), Π ∗ (t), 0 ≤ t ≤ T } : J(u∗ (·)) ≥ J(u(·)) for any u(·) : (5).
2. There exists a sequence of control functions {us (t), 0 ≤ t ≤ T } : (5) and a
value J¯ : J(us (·) → J, ¯ s → ∞, J(u(·)) ≤ J¯ for any u(·) : (5).
Proof. Let us estimate a solution of the equation (7)
Z t
w(t) = φy (t) + Gy (t − s)f (v(s), w(s)) ds ≤
0
Z t
φy (t) + Gy (t − s)f (b1 , w(s)) ds = wb1 (t).
0
Here wb1 (t) is a solution of the equation (7): v(s) ≡ b1 .
Thus, for the advertising strategy (5) the accumulated influence of previous
sales w(s) is limited by a constant value K : w(t) ≤ K = max wb1 (t).
0≤t≤T
It is possible to demonstrate that the total profit is limited:
Z T
Π(T ) = ((1 − µ)f (v(s), w(s)) − u(s)) ds ≤
0
Z T
f (v(s), w(s)) ds ≤ T max f (v, w),
0 (v,w)∈D
where D = {(v, w) : 0 ≤ w ≤ K, 0 ≤ v ≤ b1 } .
Thus, the range of the functional J(u(·)) of the optimal control problem (5),
(6), (7), (8) is limited. Denote this range as L.
Let us suppose that J¯ = sup L. Evidently, J¯ exists and it is limited.
If J¯ ∈ L, then the first alternative is realized else the second alternative is
realized [8]. t
u
Remark 2. If the second alternative of the Theorem 2 is realized, then there
exists an approximated solution of the optimal control problem (5), (6), (7),
(8) so that for any ε > 0 there exists such a control function uε (·) : (5) and
appropriate solutions of (6), (7), (8) that J¯ − J(uε (·)) < ε.
4 The maximum principle for optimization problem of
advertising costs with a Volterra integral equation
Consider the maximum principle for the problem.
Denote by x1 (t), x2 (t), x3 (t) the functions of Volterra type:
x1 (t) = v(s) − φu (s), x2 (t) = w(s) − φy (s), x3 (t) = Π(t).
And x(t) :
x1 (t) Z t Gu (t − s)u(s)
x(t) = x2 (t) = Gy (t − s)y(s) ds, (9)
x3 (t) 0 (1 − µ)(y(s) − u(s))
where y(s) = f (φu (s) + x1 (s), φy (s) + x2 (s)). Here the objective function is
x3 (T ).
Introduce the modified Hamilton-Pontryagin function:
Z T
H(s, x, u, ψ) = (ψ(s), F (s, s, x, u)) + (ψ(t), Ft0 (t, s, x, u)) dt, (10)
s
where
ψ(s) = (ψ1 (s), ψ2 (s), ψ3 (s)),
Gu (t − s)u
F (t, s, x, u) = Gy (t − s)f (φu (s) + x1 , φy (s) + x2 ) .
(1 − µ)f (φu (s) + x1 , φy (s) + x2 ) − u
Define the adjoint variables ψ(s) :
dψi (s) ∂H(s, x, u, ψ)
=− , i = 1, 2, 3, ψ1 (T ) = ψ2 (T ) = 0, ψ3 (T ) = 1. (11)
ds ∂xi
Note that the Hamiltonian function (10) is linear with respect to u. That is,
the maximum can be obtained, if only u takes the extreme values b or 0. The
partial derivative of the Hamiltonian function with respect to u is determined:
Z T
∂H(s, x, u, ψ) ∂Gu (t − s)
= ψ1 (s)Gu (0) − ψ3 (s) + ψ1 (t) dt.
∂u s ∂t
Obviously, that
∂H(s, x, u, ψ)
clb, > 0,
∂u
∂H(s, x, u, ψ)
u(s) = 0, < 0, (12)
∂u
∂H(s, x, u, ψ)
0 ≤ ū ≤ b.
ū, = 0,
∂u
It is required to solve the boundary problem (9), (11), under the condition
(12) in order to obtain a solution of the optimal control problem. Generally, the
boundary problem (9), (11) under (12) cannot be solved by analytical methods,
so we need to apply numerical schemes.
5 Practical application
In order to test the dynamic model of advertising costs, the company’s monthly
data set of advertising costs and revenue (in Russian rubles) from January 2009
to July 2014 is analyzed. The company is a producer of ready-made clothes
located in Ulyanovsk, Russia.
Consider the case when the function f (v, w) is nonlinear with respect to v
and w : f (v, w) = αv β1 wβ2 . That is, the equation (3) can be written:
Z t β1 Z t β2
y(t) = α φu (t) + Gu (t − s)u(s) ds φy (t) + Gy (t − s)y(s) ds .
0 0
Based on the properties of the functions Gu (τ ), Gy (τ ), it is possible to define
them:
Gu (τ ) = exp au τ 2 + bu τ , Gy (τ ) = exp ay τ 2 + by τ .
For example, the parameters au , bu , ay , by , β1 , β2 , α can be assessed by the
method of least squares.
Results of the experiment
Denote by T the planning period. The limited number of advertising budget
b is equal to 72,525.50 Russian rubles (maximal advertising costs in according
to the data of the last year). The assessments of parameters of the model are
obtained by the method of least squares: âu = −0.35, b̂u = 0.97, ây = 0, b̂y =
−1.37, βˆ1 = 0.16, βˆ2 = 0.93, α̂ = 1.38. The functions φu (t) and φy (t) are defined
as approximation functions for the statistical data.
The present model was analyzed by numerical scheme:
– a combination of the method of local variations and the method of successive
approximations (Krylov I.A., Chernousko F.L.) was used for solving the
optimization problem;
– the trapezoidal rule was used for calculating the integrals (6), (7), (8) (inte-
gration step h = 0.001).
The solutions are found for different planning periods: T = 1, 2 and 3 months.
In each case the solutions have the same structure:
(
b, 0 ≤ t < t∗ ,
u(t) =
0, t∗ ≤ t ≤ T,
where t∗ is a switch point of control. The necessary conditions (9), (11), (12) are
fulfilled for the solutions.
Table 1. The results of the experiment for the different planning periods T
T, months Π(T ), Russian rubles t∗ , months
1 2.90843 × 107 0.989
2 1.52341 × 108 1.994
3 6.43752 × 108 2.998
Table 1 demonstrates the obtained values of the total profit and the switch
point for the different planning periods.
As shown in Table 1, the maximum of the total profit can be achieved if the
company has the highest possible advertising costs at the beginning of the plan-
ning period. It is a logical strategy. First, companies draw attention of customers
to the goods. But later there is no need to spend money on advertising because
the previous costs continue to bring high profits. In this case, it is a waste of
money to invest in advertising at the end of the planning period.
6 Conclusion
In the practical application of the model (1), (2), (3), (4), (5), some difficulties
can appear. They are related to properties of the function f (v, w) which ensure
existence of the solution of the optimal control problem. Particularly, the mul-
tiplicative function f (v, w) = αv β1 wβ2 is continuous for any β1 > 0, β2 > 0 in
{(v, w) : v ≥ 0, w ≥ 0}, but in the case when 0 < β2 < 1, the Lipschitz condition
is not fulfilled for w = 0.
There is a special problem for a researcher: to identify the function type
f (v, w) which accords to empirical assumptions and requirements of Theorems
1 and 2. Specifically, in the practical application we have the nonlinear function
β β
f (v(t), w(t)) = α (φu (t) + v(t)) 1 (φy (t) + w(t)) 2 which is continuous and sat-
isfies Lipschitz condition with respect to w. Thus, the optimal control problem
has a solution in this case. Based on the numerical scheme, we got solutions
which satisfy necessary conditions of Pontryagin’s maximum principle, and it is
in accordance with the understanding of the optimal advertising strategy.
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