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Register a new userMarketing Mix Optimization with Practical Constraints
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Added by
Alvin Lim
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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In this paper, we address a variant of the marketing mix optimization (MMO) problem which is commonly encountered in many industries, e.g., retail and consumer packaged goods (CPG) industries. This problem requires the spend for each marketing activity, if adjusted, be changed by a non-negligible degree (minimum change) and also the total number of activities with spend change be limited (maximum number of changes). With these two additional practical requirements, the original resource allocation problem is formulated as a mixed integer nonlinear program (MINLP). Given the size of a realistic problem in the industrial setting, the state-of-the-art integer programming solvers may not be able to solve the problem to optimality in a straightforward way within a reasonable amount of time. Hence, we propose a systematic reformulation to ease the computational burden. Computational tests show significant improvements in the solution process.
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Marketing mix models (MMMs) are statistical models for measuring the effectiveness of various marketing activities such as promotion, media advertisement, etc. In this research, we propose a comprehensive marketing mix model that captures the hierarchical structure and the carryover, shape and scale effects of certain marketing activities, as well as sign restrictions on certain coefficients that are consistent with common business sense. In contrast to commonly adopted approaches in practice, which estimate parameters in a multi-stage process, the proposed approach estimates all the unknown parameters/coefficients simultaneously using a constrained maximum likelihood approach and solved with the Hamiltonian Monte Carlo algorithm. We present results on real datasets to illustrate the use of the proposed solution algorithm.
In this paper, we study a retailer price optimization problem which includes the practical constraints: maximum number of price changes and minimum amount of price change (if a change is recommended). We provide a closed-form formula for the Euclidean projection onto the feasible set defined by these two constraints, based on which a simple gradient projection algorithm is proposed to solve the price optimization problem. We study the convergence and solution quality of the proposed algorithm. We extend the base model to include upper/lower bounds on the individual product prices and solve it with some adjustments to the gradient projection algorithm. Numerical results are reported to demonstrate the performance of the proposed algorithm.
This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with a {color{black} functional or expectation} constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the constraint on devision variables. We show that this algorithm exhibits the optimal ${cal O}(1/epsilon^2)$ rate of convergence, in terms of both optimality gap and constraint violation, when the objective and constraint functions are generally convex, where $epsilon$ denotes the optimality gap and infeasibility. Moreover, we show that this rate of convergence can be improved to ${cal O}(1/epsilon)$ if the objective and constraint functions are strongly convex. We then present a variant of CSA, namely the cooperative stochastic parameter approximation (CSPA) algorithm, to deal with the situation when the constraint is defined over problem parameters and show that it exhibits similar optimal rate of convergence to CSA. It is worth noting that CSA and CSPA are primal methods which do not require the iterations on the dual space and/or the estimation on the size of the dual variables. To the best of our knowledge, this is the first time that such optimal SA methods for solving functional or expectation constrained stochastic optimization are presented in the literature.
This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We present a computable stochastic approximation type algorithm, namely the stochastic linearized proximal method of multipliers, to solve this convex stochastic optimization problem. This algorithm can be roughly viewed as a hybrid of stochastic approximation and the traditional proximal method of multipliers. Under mild conditions, we show that this algorithm exhibits $O(K^{-1/2})$ expected convergence rates for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $K$ denotes the number of iterations. Moreover, we show that, with high probability, the algorithm has $O(log(K)K^{-1/2})$ constraint violation bound and $O(log^{3/2}(K)K^{-1/2})$ objective bound. Some preliminary numerical results demonstrate the performance of the proposed algorithm.
Both Bayesian and varying coefficient models are very useful tools in practice as they can be used to model parameter heterogeneity in a generalizable way. Motivated by the need of enhancing Marketing Mix Modeling at Uber, we propose a Bayesian Time Varying Coefficient model, equipped with a hierarchical Bayesian structure. This model is different from other time varying coefficient models in the sense that the coefficients are weighted over a set of local latent variables following certain probabilistic distributions. Stochastic Variational Inference is used to approximate the posteriors of latent variables and dynamic coefficients. The proposed model also helps address many challenges faced by traditional MMM approaches. We used simulations as well as real world marketing datasets to demonstrate our model superior performance in terms of both accuracy and interpretability.
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