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:
* **$T$**: 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).
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.
This document explains how individual asset optimizations are aggregated to find the best economic strategy for the entire fleet or plant.
## 1. Fleet Aggregation Theory
At the plant level, the objective is to minimize the **Total Fleet Cost per Unit Time**. This assumes a **Uniform Interval Policy** or a **Synchronized Overhaul Strategy** where the plant looks for a common maintenance rhythm.
### Plant Objective Function
The plant-level cost function is the summation of individual equipment cost functions ($C_i$):
* **$N$**: Total number of equipments in the scope.
* **$C_{p,i}$**: Preventive cost for equipment $i$.
* **$C_{f,i}$**: Failure cost for equipment $i$.
* **$E[N_i(T)]$**: Expected failures for equipment $i$ until time $T$.
## 2. Searching for the Fleet Optimum
The "actual theory" used by the engine involves a two-phase search:
### Phase 1: Unconstrained Summation
The system calculates the CPUT curve for every piece of equipment independently. It then sums these curves to create a "Fleet U-Curve." The minimum of this sum represents the **Theoretical Fleet Optimum**.
### Phase 2: Sparepart Interaction & Constraints
Unlike a simple sum, the real plant optimum must account for **shared resources** (e.g., a limited budget or limited spare parts).
1. **Sparepart Conflicts**: If multiple equipments reach their optimal interval at the same time, the `SparepartManager` checks if there are enough parts in the warehouse.
2. **Constraint Penalty**: If parts are missing, a "Procurement Penalty" is added to the $C_{p,i}$ for that specific month, effectively shifting the "U-curve" to the right (delaying) or left (earlier) depending on availability.
3. **Final Selection**: The system chooses the Month $T$ that minimizes the *constrained* total fleet cost.
## 3. Implementation in `service.py`
The code uses `numpy` to perform vector addition of the cost curves:
* **Fleet CPUT**: The average monthly budget required to maintain the plant at the chosen interval.
* **Accumulation Control**: By using amortized costs (Rp/Month) instead of total cumulative costs, the plant chart remains stable and allows for direct comparison between intervals regardless of the number of assets.
* **Risk vs. Cost**: The plant chart shows the trade-off between the *Fleet Failure Risk* (increasing line) and *Fixed Overhaul Costs* (decreasing line).
