product life cycle; diffusion of innovations; Bass model; advertising; optimal control; dynamic programming; contraction mapping


The subject of the research is the development of a strategy for optimal management of the company's advertising expenditures when promoting a new product to the market throughout its entire life cycle. To model the sales dynamics of a new product, the article proposes a modification of the Bass diffusion of innovations model, which makes it possible to numerically simulate the impact of advertising expenditures on the intensity of sales of the two groups of consumers identified in the model, namely innovators and imitators. The problem of optimal control of advertising expenditures for product promotion throughout its entire life cycle is formulated. As an optimality criterion, the expected net present value of cash flows from the sale of the product is chosen. It is shown how to solve this problem by means of dynamic programming techniques. An algorithm for numerical search for the optimal control program of advertising expenditures based on the method of contraction mapping has been developed and implemented. As a result of the algorithm, the optimal level of advertising expenditures is established in the form of a law with feedback, where the degree of market saturation acts as a state variable of the system. On this basis, one can find the optimal level of advertising expenditures in each time period and compare sales trajectories with different advertising strategies. Two cases of advertising aimed at innovators and imitators were studied separately. Numerical experiments were carried out, on the basis of which the qualitative characteristics of the strategy for optimal control of advertising expenditures for these two cases were determined. The influence of the chosen management strategy on the form and duration of the product life cycle has been studied. Conclusions are drawn about the comparative economic efficiency of various advertising management strategies. Optimization of the advertising budget using the developed methodology allows enterprises to check whether the planned level of advertising spending is excessive or insufficient. The application of the methodology allows not only to find the optimal size of advertising expenditures depending on the stage of the product life cycle, but also to predict sales and profits for future periods.

Author Biography

Oleg Melnikov, National Technical University "Kharkiv Polytechnic Institute"

Candidate of Economic Sciences, Associate Professor


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