## Application of Derivatives to Nonlinear Programming for Prescriptive Analytics

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Gone are the days when business analytics would bank on statistics alone. Besides the traditional probability theory and statistics, the machine learning techniques of the present era, work in complete sync with linear algebra, graph theory, dynamic programming, multivariate calculus etc. As far as multivariate calculus is concerned, the different methods that lend support to machine learning algorithms are differential and integral calculus, partial derivatives, gradient and directional derivative, vector-valued function, Jacobian matrix and determinant, Hessian matrix, Laplacian and Lagrangian distributions etc. The present article will discuss the applications of second order derivatives and partial derivatives on optimization problems, as required for prescriptive analytics.

Prescriptive analytics provides precise decisions on the course of action that the business will undertake for success in future. One of the prominent applications of prescriptive analytics in marketing is the optimization problem of marketing budget allocation. The business problem is to figure out the optimum quantity of budget that needs to be allocated from the total advertising budget to each of the advertising media like TV, press, internet video etc. for maximizing the revenue. The budget optimization problem is solved either through Linear or Nonlinear Programing (NLP) which depends on whether

• The objective function is linear/ nonlinear
• The feasible region is determined by linear/nonlin­ear constraints.

Thus, one of the important assumptions for linear programming is the constant returns to scale for each of the advertising media. But the real world TV advertisement data, as plotted in Figure 1, defies such assumption as the graph shows a concave function. The constraint that may be considered for developing such optimization problem is the maximum amount that should be spent on a particular media such that beyond that point any further expenditure may lead to the increase in revenue but at a decreasing rate. Thus, it’s important to find out the diminishing point of return for each of the advertising medium. Figure 1 provided below shows the revenue generated against the cost incurred for TV advertisement. Both the cost and the revenue mentioned in the current paper are dollar value in thousands.

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