Mathematical Reference

This document provides the mathematical foundations for the Risk Simulation Framework, including reliability theory, Monte Carlo methods, and risk calculations.

1. Reliability Theory

1.1 Failure Rate (Hazard Function)

The failure rate $\lambda(t)$ is the conditional probability of failure per unit time:

\[\lambda(t) = \lim_{\Delta t \to 0} \frac{P(t < T \leq t + \Delta t | T > t)}{\Delta t} = \frac{f(t)}{R(t)}\]

For constant failure rate (exponential distribution):

\[\lambda(t) = \lambda \quad \text{(constant)}\]

1.2 Reliability Function

The reliability function $R(t)$ is the probability of survival to time $t$:

\[R(t) = P(T > t) = e^{-\int_0^t \lambda(u) du}\]

For constant failure rate:

\[R(t) = e^{-\lambda t}\]

1.3 Mean Time To Failure (MTTF)

\[\text{MTTF} = E[T] = \int_0^\infty R(t) dt = \int_0^\infty e^{-\lambda t} dt = \frac{1}{\lambda}\]

1.4 Mean Time Between Failures (MTBF)

\[\text{MTBF} = \text{MTTF} + \text{MTTR}\]

1.5 Availability

Steady-state availability:

\[A = \frac{\text{MTTF}}{\text{MTTF} + \text{MTTR}} = \frac{\text{Uptime}}{\text{Total Time}}\]

Instantaneous availability (for repairable systems):

\[A(t) = \frac{\mu}{\lambda + \mu} + \frac{\lambda}{\lambda + \mu} e^{-(\lambda + \mu)t}\]

Where $\mu = 1/\text{MTTR}$ is the repair rate.

2. System Reliability

2.1 Series System

All components must function (AND logic):

┌───┐   ┌───┐   ┌───┐
│ A │───│ B │───│ C │
└───┘   └───┘   └───┘

\[R_s(t) = \prod_{i=1}^n R_i(t)\] \[A_s = \prod_{i=1}^n A_i\]

Example: Three components with 99% availability each: $A_s = 0.99^3 = 0.970$

2.2 Parallel System (Active Redundancy)

System works if any component functions (OR logic):

    ┌───┐
┌───│ A │───┐
│   └───┘   │
│   ┌───┐   │
├───│ B │───┤
│   └───┘   │
└───────────┘

\[R_p(t) = 1 - \prod_{i=1}^n (1 - R_i(t))\] \[A_p = 1 - \prod_{i=1}^n (1 - A_i)\]

Example: Two components with 99% availability each: $A_p = 1 - (1-0.99)^2 = 0.9999$

2.3 k-out-of-n System

System works if at least $k$ of $n$ components function:

\[R_{k/n}(t) = \sum_{i=k}^n \binom{n}{i} R(t)^i (1-R(t))^{n-i}\]

2-out-of-3 system: $R_{2/3} = 3R^2(1-R) + R^3 = 3R^2 - 2R^3$

3. Monte Carlo Simulation

3.1 Random Variate Generation

Exponential distribution (for failure/repair times):

\[T = -\frac{1}{\lambda} \ln(U), \quad U \sim \text{Uniform}(0,1)\]

Weibull distribution (for wear-out failures):

\[T = \eta \cdot (-\ln(U))^{1/\beta}\]

Where $\eta$ is scale parameter, $\beta$ is shape parameter.

3.2 Simulation Algorithm

For each iteration i = 1 to N:
    t = 0
    Initialize all equipment to OPERATING
    production[i] = 0
    
    While t < T_horizon:
        # Generate next event
        For each equipment j:
            If operating: t_fail[j] = t + Exp(λ_j)
            If failed: t_repair[j] = t + Exp(μ_j)
        
        t_next = min(all event times)
        
        # Advance time and update state
        production[i] += P(state) × (t_next - t)
        t = t_next
        Update equipment states
    
    Store production[i]

Calculate statistics from production[]

3.3 Confidence Intervals

For mean:

\[\bar{X} \pm z_{\alpha/2} \frac{s}{\sqrt{n}}\]

95% confidence interval ($z_{0.025} = 1.96$):

\[CI_{95\%} = \bar{X} \pm 1.96 \frac{s}{\sqrt{n}}\]

3.4 Percentile Estimation

Order statistics method:

Sort $n$ samples: $X_{(1)} \leq X_{(2)} \leq … \leq X_{(n)}$

For percentile $p$: $\hat{X}_p = X_{(\lceil np \rceil)}$

P10, P50, P90:

P10: 10th percentile (optimistic)
P50: 50th percentile (median)
P90: 90th percentile (conservative)

3.5 Convergence

Standard error of mean:

\[SE = \frac{\sigma}{\sqrt{n}}\]

Required sample size for precision $\epsilon$:

\[n = \left(\frac{z_{\alpha/2} \cdot \sigma}{\epsilon}\right)^2\]

4. Risk Calculations

4.1 Risk Score

\[\text{Risk Score} = P \times C\]

Where:

$P$ = Probability level (1-5)
$C$ = Consequence level (1-5)

4.2 Annual Risk Cost

\[C_{\text{annual}} = \lambda \times C_{\text{event}}\]

Where $C_{\text{event}}$ is the cost per failure event.

4.3 Event Cost Components

\[C_{\text{event}} = C_{\text{production}} + C_{\text{downtime}} + C_{\text{repair}}\]

Production loss cost: $C_{\text{production}} = \text{MTTR} \times \dot{m} \times \text{Loss\%} \times P_{\text{product}}$

Where:

$\dot{m}$ = mass flow rate (kg/hr)
$P_{\text{product}}$ = product price ($/kg)

Downtime cost: $C_{\text{downtime}} = \text{MTTR} \times C_{\text{fixed}}$

4.4 Expected Annual Production Loss

\[E[\text{Loss}] = \sum_{i=1}^n \lambda_i \times \text{MTTR}_i \times L_i\]

Where $L_i$ is the production loss fraction for equipment $i$.

4.5 Production Availability

\[A_{\text{production}} = 1 - \sum_{i=1}^n \frac{\lambda_i \times \text{MTTR}_i \times L_i}{8760}\]

5. Production Impact

5.1 Production Loss Percentage

\[L\% = \frac{P_{\text{normal}} - P_{\text{degraded}}}{P_{\text{normal}}} \times 100\%\]

5.2 Capacity Factor Effect

When equipment operates at reduced capacity $C_f$:

\[P_{\text{degraded}} = P_{\text{normal}} \times f(C_f)\]

For simple proportional relationship: $f(C_f) = C_f$

For non-linear (e.g., compressor at reduced speed): $f(C_f) = C_f^\alpha, \quad \alpha > 1$

5.3 Criticality Index

\[CI_i = \frac{L_i}{\max_j(L_j)}\]

Equipment with $CI > 0.8$ is “critical”.

6. Dependency Analysis

6.1 Cascade Effect Propagation

For equipment $j$ downstream of failed equipment $i$:

\[L_j = L_i \times T_{ij}\]

Where $T_{ij}$ is the transmission factor (0-1).

6.2 Criticality Increase

When equipment $i$ fails, criticality of parallel equipment $j$ increases:

\[CI_j^{\text{new}} = CI_j^{\text{base}} \times \frac{n}{n - 1}\]

Where $n$ is the number of parallel trains.

6.3 Cross-Installation Impact

\[L_{\text{target}} = L_{\text{source}} \times \text{Impact Factor}\]

7. Probability Distributions

7.1 Exponential Distribution

PDF: $f(t) = \lambda e^{-\lambda t}$

CDF: $F(t) = 1 - e^{-\lambda t}$

Mean: $E[T] = 1/\lambda$

Variance: $\text{Var}[T] = 1/\lambda^2$

7.2 Weibull Distribution

PDF: $f(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1} e^{-(t/\eta)^\beta}$

Mean: $E[T] = \eta \cdot \Gamma(1 + 1/\beta)$

Special cases:

$\beta = 1$: Exponential (constant failure rate)
$\beta < 1$: Decreasing failure rate (infant mortality)
$\beta > 1$: Increasing failure rate (wear-out)

7.3 Log-Normal Distribution

For repair times:

PDF: $f(t) = \frac{1}{t\sigma\sqrt{2\pi}} e^{-\frac{(\ln t - \mu)^2}{2\sigma^2}}$

Mean: $E[T] = e^{\mu + \sigma^2/2}$

8. Statistical Formulas

8.1 Sample Mean

\[\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i\]

8.2 Sample Variance

\[s^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2\]

8.3 Coefficient of Variation

\[CV = \frac{s}{\bar{X}} \times 100\%\]

8.4 Binomial Coefficient

\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

9. Numerical Methods

9.1 Newton-Raphson (for optimization)

\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

9.2 Trapezoidal Integration

\[\int_a^b f(x)dx \approx \frac{h}{2}\sum_{i=1}^{n-1}(f(x_i) + f(x_{i+1}))\]

9.3 Linear Interpolation (for percentiles)

\[X_p = X_k + (p \cdot n - k)(X_{k+1} - X_k)\]

10. Unit Conversions

From	To	Factor
hours	years	÷ 8760
failures/year	failures/hour	÷ 8760
kg/hr	tonnes/day	× 0.024
bara	psia	× 14.5038
°C	K	+ 273.15