# Equipment Level Overhaul Optimization

This document explains the mathematical theory and implementation used to determine the optimal overhaul interval for individual assets.

## 1. Theoretical Foundation
The optimization model is based on the **Expected Total Cost per Unit Time (CPUT)** theory, a standard approach in reliability engineering for age-replacement and block-replacement policies.

### Objective Function
The goal is to find an overhaul interval ($T$) that minimizes the total expected cost amortized over time:

$$C(T) = \frac{C_{Preventive} + C_{Corrective} \cdot E[N(T)]}{T}$$

Where:
*   **$T$**: The overhaul interval (e.g., month 12, 24, etc.).
*   **$C_{Preventive}$**: The cost of a planned overhaul (materials, labor, and procurement).
*   **$C_{Corrective}$**: The cost of an unplanned failure (repairs + risk cost of downtime).
*   **$E[N(T)]$**: The expected number of failures occurring in the interval $(0, T]$. This is derived from the **NHPP (Non-Homogeneous Poisson Process)** reliability model.

## 2. The Cost Balance (The "U-Curve")
Optimization works by balancing two competing costs:
1.  **Overhaul Cost ($C_{PM}/T$)**: As the interval $T$ increases, the amortized cost of the overhaul decreases (economy of waiting).
2.  **Failure Risk ($C_{CM} \cdot E[N(T)]/T$)**: As the interval $T$ increases, the probability and expected frequency of failure increase (cost of wear-out).

The point where these two lines intersect typically represents the **Global Minimum** of the total cost curve, known as the **Optimum Overhaul Month**.

## 3. Implementation Logic
In the `service.py` engine, the search is performed as follows:

1.  **Failure Projection**: The system fetches reliability prediction data (cumulative failures) for the analysis window.
2.  **Sparepart Simulation**: For every potential month $T$, the system simulates the procurement process to calculate the real $C_{Preventive}$ (including any shortage penalties).
3.  **Cost Amortization**:
    ```python
    total_cycle_cost = total_expected_failure_cost + total_preventive_cost
    cput = total_cycle_cost / month_index
    ```
4.  **Grid Search**: The system iterates through all months in the analysis window and identifies the index with the lowest `cput` value.

## 4. Interpretation in UI
*   **Analysis Window Card**: Shows the CPUT value at your currently selected month.
*   **Optimum Target Card**: Shows the month $T$ where the curve is at its lowest point.
*   **Potential Benefit**: Calculated as $CPUT(current) - CPUT(optimum)$. This represents the real monthly cash-flow saving if the maintenance interval is adjusted.