Explainable Multi Objective Optimization of Digital Marketing Budget Allocation Under Uncertain Consumer Response

I. Introduction

Figure 1. EMOO Framework Architecture.

Figure 1. EMOO Framework Architecture.

II. Related Work

III. Methodology

A. Research Design

The study employs a quantitative, ex post facto research design using a simulated marketing performance dataset representing five industry sectors over 24 months. The operational workflow proceeds sequentially through five stages: data acquisition and preprocessing, Bayesian hierarchical model specification and fitting, posterior predictive simulation for uncertainty propagation, NSGA II multi objective optimization, and SHAP based explanation generation. Each stage produces structured outputs consumed by the subsequent stage, forming a reproducible pipeline from the simulated channel level data to interpretable budget allocation recommendations. The design emphasizes transparency, with all model specifications, hyperparameters, and convergence diagnostics documented to support independent replication.

Figure 2. Research Design Operational Workflow.

Figure 2. Research Design Operational Workflow.

Figure 2 presents the five stage research design workflow. Stage 1 prepares the simulated marketing data and produces a clean channel week dataset. Stage 2 specifies and fits the Bayesian hierarchical response model. Stage 3 generates posterior predictive simulations to represent response uncertainty. Stage 4 applies NSGA II to produce a Pareto

D. Analytical Model and System Architecture

The Bayesian hierarchical model specifies channel effects as:

$$y_t\sim \mathrm{Normal}\left({\mu }_t\right|\sigma )$$

$${\mu }_t={\gamma }_0+{\displaystyle \sum _{c=1}^6}{\gamma }_c\cdot S_{c,t}+{\delta }_{\sec \mathrm{tor}\left[t\right]}$$

$${\gamma }_c\sim \mathrm{Normal}\left({\mu }_{\gamma }\right|{\tau }_{\gamma })$$

$${\delta }_k\sim \mathrm{Normal}\left(0\right|{\tau }_{\delta })$$

Where, $y_t$is revenue at week $t$, ${\gamma }_c$represents channel effects, ${\delta }_k$captures sector-level random effects, and priors are weakly informative. Posterior sampling uses Hamiltonian Monte Carlo with 4 chains, 2,000 warmup iterations, and 2,000 sampling iterations.

The NSGA-II optimizer maximizes three objectives:

$$f_1\left(w\right)=\mathbb{E}\left[{\displaystyle \sum _{c=1}^6}{\widehat{y}}_c\left(w_c\right)\right](\mathrm{Expected}\mathrm{Revenue})$$

$$f_2\left(w\right)=\mathrm{Var}\left[{\displaystyle \sum _{c=1}^6}{\widehat{y}}_c\left(w_c\right)\right](\mathrm{Allocation}\mathrm{Risk})$$

$$f_3\left(w\right)={\displaystyle \sum _{c=1}^6}{w}_c\cdot {\mathrm{BrandWeight}}_c(\mathrm{Brand}\mathrm{Presence})$$

ubject to $\sum w_c=B$(total budget) and $w_c\ge 0$. Population size is 200, generations are 100, crossover probability is 0.9, and mutation probability is 0.1.

Table 1. Data Flow and Parameter Specifications.

Table 1. Data Flow and Parameter Specifications.

Component	Input	Output	Key Parameters
Bayesian Model	Spend, Impressions, Conversions, Revenue	Posterior distributions for ${\gamma }_c$, ${\lambda }_c$, ${\beta }_c$	Priors: ${\mu }_{\gamma }\sim N\left(0\right|10)$, ${\tau }_{\gamma }\sim \mathrm{HalfCauchy}\left(5\right)$
NSGA-II	Posterior predictive samples, Budget constraint	Pareto frontier of allocations ${\mathit{w}}^{*}$	Pop: 200, Gen: 100, $p_c$: 0.9, $p_m$: 0.1
SHAP	Pareto allocations, Fitted model	Channel contributions, Reallocation rules	KernelSHAP samples: 1,000, Background: 100

E. Validation Checks and Optimization

Model validation employs posterior predictive checks, comparing simulated revenue distributions against observed data. Convergence diagnostics include $\widehat{R}$statistics (target < 1.05) and effective sample size (target > 400). NSGA-II performance is assessed via hypervolume indicator and spacing metric to ensure frontier diversity. Optimization terminates when hypervolume improvement falls below 0.001 over 10 consecutive generations or maximum generations (100) are reached.

Figure 3. EMOO System Architecture.

Figure 3. EMOO System Architecture.

IV. Results and Discussion

A. Results

This section presents the evaluation results obtained from applying the Explainable Multi Objective Optimization (EMOO) framework to the 24 month simulated multi channel marketing dataset representing five industry sectors. The results are organized into four subsections: model fit and convergence diagnostics, Pareto frontier characterization, baseline versus proposed allocation performance, and SHAP based explanation analysis. Each subsection reports quantitative metrics and graphical results from the simulation based evaluation.

The Bayesian hierarchical response model demonstrated strong predictive accuracy and parameter convergence. Posterior predictive checks yielded an $R^2$value of 0.91, indicating that 91% of variance in weekly revenue was explained by the model after accounting for adstock transformation, saturation effects, and sector-level random intercepts. Root mean squared error (RMSE) on normalized revenue was 0.18, corresponding to an average prediction error of approximately $180 per $1,000 of revenue. Convergence diagnostics confirmed reliable posterior estimation. All $\widehat{R}$statistics (Gelman-Rubin diagnostic) fell below 1.02 across 24 monitored parameters, well under the 1.05 threshold recommended for convergence. Effective sample sizes exceeded 400 for all channel effect parameters (${\gamma }_c$), ensuring stable estimates of posterior means and credible intervals. Channel effect posteriors revealed heterogeneity in both magnitude and uncertainty. Paid Search exhibited the largest mean effect ($\gamma =0.42$) with relatively narrow 95% credible interval width (0.16), reflecting consistent high ROI. Email followed closely ($\gamma =0.35$, CI width = 0.18), while Display showed the smallest mean effect ($\gamma =0.19$) and widest uncertainty (CI width = 0.30), suggesting volatile performance across sectors and time periods. These posterior distributions formed the input for downstream NSGA-II optimization, with uncertainty propagated through Monte Carlo sampling of 1,000 posterior predictive draws per allocation scenario. The NSGA-II optimizer generated a Pareto frontier comprising 187 non-dominated solutions across the three objectives: expected revenue, allocation risk (variance), and brand presence score. Visual inspection of the frontier in three-dimensional objective space revealed a convex surface, indicating diminishing marginal returns as allocations shifted toward extreme strategies. Solutions clustered naturally into three groups based on objective trade-offs, which were identified using k-means clustering (k=3) on normalized objective values.

Table 2. Representative Pareto-Optimal Allocations by Strategy Cluster.

Table 2. Representative Pareto-Optimal Allocations by Strategy Cluster.

CHANNEL	REVENUE-MAXIMIZING (%)	RISK-MINIMIZING (%)	BALANCED (%)	CURRENT BASELINE (%)
PAID SEARCH	35	25	30	28
PAID SOCIAL	20	15	18	22
DISPLAY	10	20	15	18
VIDEO	15	15	15	12
EMAIL	15	20	17	15
AFFILIATE	5	5	5	5
EXPECTED REVENUE ($K)	1,247	1,089	1,198	1,012
ALLOCATION RISK (VARIANCE)	0.089	0.042	0.061	0.095
BRAND PRESENCE SCORE	0.62	0.78	0.71	0.65

The Revenue-Maximizing cluster concentrated 55% of budget on Paid Search and Paid Social, the two channels with highest posterior mean effects. This strategy achieved the highest expected revenue ($1,247K) but incurred the greatest allocation risk (variance = 0.089), reflecting exposure to saturation and competitive bidding volatility in these channels. The Risk-Minimizing cluster diversified spend more evenly, allocating 40% to Display and Email, which exhibited lower posterior variance despite smaller mean effects. This approach reduced allocation risk by 53% relative to the Revenue-Maximizing strategy but sacrificed 12.7% in expected revenue. The Balanced cluster represented a compromise, shifting 5 percentage points from Paid Social to Display relative to the Revenue-Maximizing allocation, achieving 96% of maximum revenue with 32% lower risk.

Figure 4. Pareto Frontier in Objective Space.

Figure 4. Pareto Frontier in Objective Space.

The frontier exhibits a clear trade-off surface: moving from the lower-left (low revenue, low risk) to the upper-right (high revenue, high risk) requires accepting greater variance to achieve higher expected returns. Brand presence score, represented by depth into the page, shows moderate positive correlation with revenue (r = 0.41) but negative correlation with risk (r = -0.28), indicating that brand-building channels (e.g., Video, Display) provide stability but lower immediate returns. To quantify operational impact, the Balanced Pareto solution was compared against the current baseline allocation used by participating firms. The proposed allocation increased expected revenue by $186K (18.4%) while simultaneously reducing allocation risk by 35.8%, demonstrating that multi-objective optimization can identify solutions that dominate single-objective baselines on multiple criteria. Return on advertising spend (ROAS) improved from 3.2 to 3.8, reflecting more efficient deployment of budget toward channels with higher marginal returns. The 95% credible interval width for predicted revenue narrowed from 0.34 to 0.21 (38.2% reduction), indicating that the optimized allocation not only increased mean returns but also reduced uncertainty around those returns.

Figure 5. Expected Revenue by Channel (Baseline vs. Proposed).

Figure 5. Expected Revenue by Channel (Baseline vs. Proposed).

Table 3. Baseline vs. Proposed Model Performance Metrics.

Table 3. Baseline vs. Proposed Model Performance Metrics.

Metric	Current Baseline	Proposed (Balanced)	Improvement (%)
Expected Revenue ($K)	1,012	1,198	+18.4
Allocation Risk (Variance)	0.095	0.061	-35.8
Brand Presence Score	0.65	0.71	+9.2
ROAS (Revenue/Spend)	3.2	3.8	+18.8
95% Credible Interval Width	0.34	0.21	-38.2

Paid Search shows the largest absolute revenue gain (+$52K), consistent with its 2-percentage-point allocation increase and high posterior effect size. Display revenue declines slightly (-$8K) due to reduced allocation, but this is more than offset by gains in Search and Email. The pattern confirms that reallocation toward high-ROI channels drives aggregate revenue improvement without requiring additional total budget

Figure 6. Budget Allocation Shares (Balanced Strategy).

Figure 6. Budget Allocation Shares (Balanced Strategy).

Paid Search commands the largest share (30%), followed by Paid Social (18%) and Email (17%), together accounting for 65% of total budget. Display and Video each receive 15%, reflecting their dual role in driving moderate revenue while supporting brand presence objectives. Affiliate remains at 5%, consistent with its niche role in performance marketing.

Figure 7. Cumulative Expected Revenue Over NSGA-II Generations.

Figure 7. Cumulative Expected Revenue Over NSGA-II Generations.

Expected revenue increases rapidly during the first 40 generations, gaining $156K (15.4% of final value), then plateaus as the algorithm refines the Pareto frontier. The hypervolume indicator stabilized after generation 78, with improvements below 0.001 over 10 consecutive generations, triggering early termination at generation 82. This convergence pattern indicates that NSGA-II efficiently explores the solution space within the first half of the maximum generation limit

Figure 8. Channel Allocations (Baseline vs. Proposed).

Figure 8. Channel Allocations (Baseline vs. Proposed).

This graph highlights three key shifts: Paid Search increases from 28% to 30%, Paid Social decreases from 22% to 18%, and Display decreases from 18% to 15%. Email remains stable at 15–17%, while Video and Affiliate show minimal change. The visual pattern confirms that optimization primarily reallocates between the top three channels (Search, Social, Display) while maintaining stable allocations for secondary channels. SHAP (Shapley Additive explanations) values were computed for each Pareto-optimal allocation to decompose revenue predictions into channel-level contributions. Mean absolute SHAP values across 1,000 posterior samples identified Paid Search as the dominant revenue driver (mean |SHAP| = 0.38), followed by email (0.29) and Paid Social (0.24). subconscious Display exhibited the highest SHAP variance (0.042), consistent with its wide posterior effect interval and explaining its reduced allocation in risk-minimizing strategies. Measured SHAP interaction values revealed a positive synergy between Paid Search and Email (interaction effect = +0.07, p < 0.01), indicating that coordinated campaigns across these channels produce supra-additive effects. This finding suggests that allocating budget to both channels simultaneously yields greater returns than independent optimization would predict. Negative interactions were observed between Paid Social and Display (-0.04), implying audience overlap or competitive cannibalization when both channels receive high spend. SHAP dependence plots for Paid Search showed a saturation threshold at approximately 35% budget share, beyond which marginal SHAP contributions declined sharply. This pattern aligns with the Hill function saturation parameter (${\beta }_{\mathrm{Search}}=0.33$) estimated by the Bayesian model and validates the optimization constraint preventing Search allocations above 35% in risk-aware strategies.

V. Conclusion

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