# Optimisation Process

## Problem Statement

Conversion Value Bidding seeks to determine the optimal distribution of a fixed advertising budget across various customer journey touchpoints to achieve the maximum total expected conversion value. The objective is to devise an efficient budget allocation strategy that maximizes conversion value.

## Mathematical Formulation

### Customer Journey Graph

Let $G = (V, E)$ be a Directed Acyclic Graph (DAG) representing the customer journey, where:
- $V = \{v_1, v_2, \ldots, v_n\}$ is the set of nodes representing touchpoints
- $E$ is the set of directed edges representing causal relationships

### Model Parameters

For each node $a \in V$, we define:
- $\beta_{a0}$: Baseline conversion propensity
- $\beta_{a1}$: Budget sensitivity coefficient
- $\beta_{aj}$: Parent node influence coefficient (for each parent node $j$)
- $\alpha_a$: Conversion value associated with node $a$

### Decision Variables

The budget allocation vector $x = (x_{v_1}, x_{v_2}, \ldots, x_{v_n})$, where $x_a$ denotes the budget allocated to node $a$.

### Constraints

- Total Budget Constraint: $\sum_{a \in V} x_a \leq B$
- Non-negativity Constraint: $x_a \geq 0, \forall a \in V$
- Optional Node-Specific Budget Limits: $L_a \leq x_a \leq U_a, \forall a \in V$

### Objective Function

The conversion probability at each node is modelled using:

$$P_a(x) = \sigma\left(\beta_{a0} + \beta_{a1} \ln\left(1 + \frac{x_a}{S}\right) + \sum_{j \in pa(a)} \beta_{aj} P_j(x)\right)$$

Where:
- $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function
- $S$ is a scaling factor (typically 1000)
- $\ln\left(1 + \frac{x_a}{S}\right)$ introduces diminishing returns

The objective is to maximize:

$$\max_{x} \mathbb{E}[CV(x)] = \sum_{a \in V} \alpha_a \cdot P_a(x)$$

## Optimisation Algorithm: Genetic Algorithm

Given the non-linear and potentially non-convex nature of the objective function, a Genetic Algorithm (GA) is employed:

1. **Population Initialization**: Generate an initial population of candidate budget allocation vectors
2. **Fitness Evaluation**: Evaluate each candidate using a robust conversion value function
3. **Parent Selection**: Select parent individuals using tournament selection
4. **Crossover and Mutation**: Generate new offspring through genetic operators
5. **Population Replacement**: Form a new population with generated offspring
6. **Iteration and Convergence**: Repeat the process for a predefined number of generations

## Implementation Details

### Parameter Loading

Loads essential model parameters from the `parameter_summary.csv` file generated during the Estimation stage.

### Robust Conversion Value Function

Calculates the expected conversion value for a given budget allocation, incorporating:
- Parameter uncertainty
- Parent node effects
- Diminishing returns
- Constraint penalties

### Genetic Algorithm Class

The `GeneticOptimizer` class encapsulates:
- Population management
- Fitness evaluation
- Parent selection
- Crossover operations
- Mutation
- Optimization loop

### Optimization Execution and Output

The process outputs the optimal budget allocation as a dictionary, mapping each customer journey node to its recommended budget allocation.