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 previous 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 consumer reaction on advertising does not take the specified value and tends to infinity in the long run. Also the equation of consumer demand is determined. 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 mathematical 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.
The main purpose of any commercial company is to satisfy clients, encourage them to repeat purchases and to recommend company’s products to other potential 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
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:
(
Gu( ) 0; 8 2 [0; +1); lim !+1
Gu( ) = 0; G0u( ) 0; 2 [0; u );
G0u( ) 0; 2 ( u ; +1): 3. Also we can note that consumers repeat purchases because of their positive experience. In this case, the consumer demand becomes higher and the function Gy( ) 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( ) has the properties: Gy( ) 0; 8 2 [0; +1);
Gy( ) = 0; G0y( ) 0; 2 [0; y );
G0y( )
2 ( y ; +1): lim !+1 0;
The financial result of the company is the profit or loss which is determined by the condition (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: 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 . We have following: u(t) b; t 2 [0; T ]: (4)
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 (
Rewrite (
Denote by u(t); y(t) the following functions: The functions v(t); w(t);
(t) can be represented: Gu(t s)u(s) ds;
Thus, the maximization of (T ) under conditions (5), (6), (7), (8) is
equivalent to the maximization of (4) under conditions (5), (
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( ) is continuous, then v(t) is continuous in [0; T ]: Considering (5), (6) we can state the existence of the value b1 > 0 : 0
Thus, the function of the accumulated advertising effect satisfies the condition 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( ) 2 C([0; T ]); Gy( ) 2 C([0; T ]); the function 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 2 [0; T ] that satisfies the restriction (5), there exists continuous and unique function w(t); t 2 [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 : jf (v; w1) f (v; w2)j Ljw1 w2j; 8w1; w2: Define the operator A in the following form: (M Lt)k Evidently, there exists k : < 1. It means, that operator Ak is contractk! ing. Thus, the solution w(t) of the equation (7) exists, it is unique and continuous in [0; T ].
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 nondecreasing monotonously with respect to w: If the finite partial derivative f 0 w 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 (T ) under the conditions (5), (6), (7), (8).
A finite value of the accumulated profit (T ) is a functional of control u( ): Introduce J (u( )) (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 fu (t); 0 t T g : (5), a solution to the equations (6), (7), (8) fv (t); w (t); (t); 0 t T g : J (u ( )) J (u( )) for any u( ) : (5).
2. There exists a sequence of control functions fus(t); 0 t T g : (5) and a value J : J (us( ) ! J ; s ! 1; J (u( )) J for any u( ) : (5).
Proof. Let us estimate a solution of the equation (7) where D = f(v; w) : 0 w K; 0 v b1g :
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 2 L; then the first alternative is realized else the second alternative is realized [8]. 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 " > 0 there exists such a control function u"( ) : (5) and appropriate solutions of (6), (7), (8) that J J (u"( )) < ". 4
Consider the maximum principle for the problem.
Denote by x1(t); x2(t); x3(t) the functions of Volterra type: x1(t) = v(s) u(s); x2(t) = w(s) y(s); x3(t) = (t): And x(t) :
0 x1(t) 1 x(t) = @ x2(t) A = x3(t)
Z t 0 where y(s) = f ( u(s) + x1(s); y(s) + x2(s)): Here the objective function is x3(T ):
Introduce the modified Hamilton-Pontryagin function:
H(s; x; u; ) = ( (s); F (s; s; x; u)) + ( (t); Ft0(t; s; x; u)) dt; (10) where
(s) = ( 1(s); 2(s); 3(s)); s)u s)f ( u(s) + x1; y(s) + x2) )f ( u(s) + x1; y(s) + x2) u 1
A : ; i = 1; 2; 3; 1(T ) = 2(T ) = 0; 3(T ) = 1: (11) Define the adjoint variables (s) : d i(s) ds = Obviously, that
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: = 1(s)Gu(0) 3(s) + 1(t)
dt: Z T s Z T
s > 0; < 0; 0 u b: (12)
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
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) = v 1 w 2 : That is, the equation (
Z t
0
where t is a switch point of control. The necessary conditions (9), (11), (12) are
fulfilled for the solutions.
In the practical application of the model (
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
f (v(t); w(t)) = ( u(t) + v(t)) 1 ( y(t) + w(t)) 2 which is continuous and
satisfies 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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