Gap = Nominal gap +/- sum of all tolerances

For a simple assembly:
Gap_min = Nominal - sum(all positive contributors)
Gap_max = Nominal + sum(all negative contributors)

Or using sensitivity:
Gap = sum(ai * xi)
Tolerance = sum(|ai| * ti)

Where:
ai = sensitivity coefficient (+1 or -1)
xi = nominal dimension
ti = tolerance on dimension i

Define positive direction:
- Dimensions adding to gap: positive (+1)
- Dimensions subtracting from gap: negative (-1)

Example (shaft in hole):
Gap = Hole_dia - Shaft_dia
Hole: +1 (increases gap)
Shaft: -1 (decreases gap)

Statistical tolerance (RSS):
T_rss = sqrt(sum(ti^2))

For unequal distributions (weighted):
T_rss = sqrt(sum((ai * ti)^2))

Assumes:
- Normal distribution
- Independent variables
- Process centered at nominal

Cp = (USL - LSL) / (6 * sigma)
Cpk = min((USL - mean)/(3*sigma), (mean - LSL)/(3*sigma))

For 6-sigma quality:
Cpk >= 2.0
PPM defective < 3.4

For tolerance analysis:
sigma = T / (3 * k)

Where k depends on desired Cpk:
k = 3 for Cpk = 1.0
k = 4 for Cpk = 1.33
k = 6 for Cpk = 2.0

1. Define distribution for each dimension
   - Normal: mean, sigma
   - Uniform: min, max
   - Skewed: appropriate parameters

2. Generate random samples (N = 10,000+)
3. Calculate assembly result for each sample
4. Analyze output distribution
5. Determine percent out-of-spec

Position tolerance contribution:
Dia_positional / 2 = linear contribution (per direction)

For MMC position:
Contribution = (Position_tol + Bonus_tol) / 2

Bonus tolerance:
Bonus = |Actual_size - MMC_size|

Stack-up must follow datum precedence:
1. Establish primary datum (constrains normal)
2. Establish secondary datum (constrains one rotation)
3. Establish tertiary datum (constrains remaining DOF)

Feature control frame specifies:
|Position|0.5 MMC|A|B|C|

If tolerance too tight:
1. Increase gap nominal (if possible)
2. Tighten critical dimension tolerances
3. Add adjustment or shim
4. Change assembly method
5. Accept higher defect rate

If tolerance too loose:
1. Relax non-critical tolerances
2. Reduce manufacturing cost

Approximate relationship:
Cost ~ 1 / Tolerance^n

Where n ~ 1.5 to 2 for machining

Tighten tolerances on:
- Lower cost features
- Higher sensitivity contributors

{
  "analysis_name": "string",
  "requirement": {
    "type": "gap|clearance|interference|alignment",
    "nominal": "number",
    "min": "number",
    "max": "number"
  },
  "contributors": [
    {
      "name": "string",
      "nominal": "number",
      "tolerance": "number (bilateral half)",
      "direction": "+1|-1",
      "distribution": "normal|uniform",
      "cpk": "number (if normal)"
    }
  ],
  "method": "worst_case|rss|monte_carlo|all"
}

{
  "analysis_summary": {
    "requirement": {
      "min": "number",
      "max": "number"
    },
    "nominal_result": "number"
  },
  "worst_case": {
    "min_result": "number",
    "max_result": "number",
    "pass_fail": "pass|fail",
    "margin": "number"
  },
  "statistical": {
    "mean": "number",
    "sigma": "number",
    "min_3sigma": "number",
    "max_3sigma": "number",
    "percent_out_of_spec": "number",
    "cpk": "number"
  },
  "monte_carlo": {
    "mean": "number",
    "sigma": "number",
    "min_observed": "number",
    "max_observed": "number",
    "percent_out_of_spec": "number",
    "histogram": "data reference"
  },
  "sensitivity_ranking": [
    {
      "contributor": "string",
      "sensitivity": "number",
      "percent_contribution": "number"
    }
  ],
  "recommendations": "array"
}