A stochastic nonlinear programming model for budget mix optimization of digital marketing campaigns under uncertainty

Abstract This study introduces a stochastic mixed-integer nonlinear programming (MINLP) model designed to optimize budget allocation for digital marketing campaigns under uncertainty. The model maximizes Expected Net Reach (ENR) by integrating the Sainsbury Normal Method (SNM) to correct for audience duplication across platforms and employing Sample Average Approximation (SAA) to handle volatility in key metrics like impressions and reach. The primary contribution is a novel formulation that combines stochastic programming with net reach estimation, a methodological advance that avoid duplicated impressions and inaccurate aggregated metrics. This study also incorporates a weighting mechanism based on target audience demographics, enabling flexible allocation across digital platforms. To test and validate the proposed model, we conducted four case studies across various industries (Banking, Entertainment, Education, and FMCG) that demonstrated significant improvements in impressions and reach. Experimental analyses were performed to assess the performance of four budget allocation mechanisms—Empirical Weighting, Uniform Allocation, Meta-Focused, and Meta-Minimized—across different budget scenarios (50%, 100%, and 150% of the nominal budget). The numerical results highlight the dominance of Meta and YouTube platforms in brand awareness campaigns, consistently yielding the highest return on impressions. For example, in the entertainment sector, YouTube generated 77% of total impressions with only 56% of the budget, outperforming Meta and TikTok. Similarly, in the banking sector, Meta achieved 75% of impressions with a 39% budget share, proving to be the most efficient platform. These findings provide actionable insights for marketers seeking to optimize budget allocation in digital campaigns, offering a structured and adaptive framework to navigate the complexities of modern advertising.