Management based on the application of operation research methods

x_jf - sales volume of product j on segment f; where j ∈ J1 and j ∈ J2,
where J1 is the set of products j with which the company is already working in the market;
J2 - the set of products j, according to which the company must make a decision on entering the market;
f ∈ F, F - the set of segments f on which the enterprise can work with its products;
w_jf - Boolean variables that control the inclusion plan of production and sale of "new", previously developed products; moreover, the variables w_jf are such that w_jf = 1 if the product j is to be sold on the market segment f and wjf = 0 otherwise;
g_jf - unit price of product j at which the product will be sold on segment f;
k_jf - Boolean variables that track the fact that the calculated prices k_jf exceed the specified marginal prices q_jf^пр for segment f;

The variables k_jf are such that k_jf = 1, if g_jf ≤ q_jf^пр and k_jf = 0, if g_jf > q_jf^пр.

The system restrictions on the values of controlled variables describes the operating conditions that investigated production system.
The limitation that guarantees the enterprise achievement in the planned period of a given level profit from the production and sale products has the form:
,
where S_jf is the cost of selling unit product j on segment f;
- unit price of the resource l;
m_jl is the rate of consumption resource l per unit of product j;
L is the set of items production resources l;
P₀– the desired value of the company's profit from product sales for the planning period;

The restrictions guaranteeing the company to achieve a given value market share for each segment under study are reduced to a system of inequalities form:

where b_f is the desired value of the market share of the f-th segment;
E_f is the market capacity of segment f.
Restrictions on the values volume product supply in various market segments are:



,
where a_jf, ā_jf are the lower and upper bounds of the supply volume product j in the f segment (a_jf, is the mandatory part of the production volume product j to sell it in the f segment; ā_jf is the solvent demand for product j in the f segment).
Restrictions on production resources, guaranteeing that the estimated resource requirements do not exceed the levels of available resources, are reduced to the following inequalities:

where M_l is the level of available resources that type l in the planning period. Restrictions on product price values are written as follows:

where q_jf is the lower bound on the unit price of product j on segment f (for example, the cost of the product).
q_jf^пр – marginal price of product j on segment f.
The optimality criterion (objective function) of the task is to maximize the expected profit from product sales for the planning period:

As a result, we obtain the following statement of the problem: it is required to find such values ​​ x^*=||x_jf^*||, w^*=||w_jf^*||, g^*=||q_jf^*|| и k^*=||k_jf^*|| controlled variables that would satisfy the constraint system and at the same time deliver the maximum objective function P (x, w, q, k).
The optimization model belongs to the class of nonlinear programming models with controlled variables integer (Boolean) and continuous type [3, 4].
To analyze the model, it is proposed to use a heuristic algorithm based on a phased increase in the prices of products and the solution at each stage corresponding problem to the partially integer programming using the branch and bound method (Land and Doig method). With an iterative increase in prices (starting from the cost of products), the expected profit should initially grow due to the growth in revenue. In the future, certain types of products for which the current price values ​​will exceed the marginal prices for the segments will begin to “drop out” of the segments. As a result, profit growth should slow down and starting a certain iteration, profit will decrease. The values ​​of supply volumes and product prices, as well as the many remaining segments at a certain step iterative process, at which the maximum profit enterprise is achieved, will correspond to the optimal solution to the problem.
As a result of solving the problem, it seems possible to optimize the choice of target segments, assortment and sales volumes products, as well as product prices in each segment; most fully take into account consumer demand; maximize the expected profit from product sales and the efficiency of using production resources.


The solution to the problem on the PC in relation to the communications industry.

Initial data. The distribution has already developed and alternative types of services by market segments are presented in Table 1. The symbol "*" means the possibility of the communications enterprise working with this service in the market segment, and the symbol "**" - a new (alternative) service.
TABLE 1 Distribution of services by market segment

Service Code	Service	Segment Code