1 Introduction
1.1 Background
1.2 PRA in Fire Safety Engineering
2 Development of the J-Value
2.1 Cost Optimisation in Cost–Benefit Analysis
2.2 The Life Quality Index (LQI)
2.3 LQI and J-Value
2.3.1 Societal Willingness to Pay
Country | SWTP π—life risk reduction is associated with a proportional change in mortality across the age distribution | SWTP ∆—change in mortality due to life risk reduction is uniformly distributed across the age distribution | ||||
---|---|---|---|---|---|---|
2% | 3% | 4% | 2% | 3% | 4% | |
UK | 2600 | 2178 | 1873 | 4105 | 3665 | 3270 |
USA | 2488 | 2100 | 1822 | 3187 | 2833 | 2542 |
Australia | 3061 | 2614 | 2279 | 4840 | 4298 | 3843 |
India | 128 | 110 | 93 | 175 | 156 | 139 |
2.3.2 Derivation of a Fire Safety Variant of the J-Value
3 Inclusion of Other Fire Safety Benefits and Discounting
3.1 Injury and Damage
3.2 Renewal Process and Implications for Valuation of Damages
3.3 Inclusion of Discounting
3.4 Implementation and Continuing Costs
3.5 Application of the J-Value to an Exemplar Scheme
Input | Symbol | Metric | Unit |
---|---|---|---|
Number of buildings that could be affected by fire |
N
units
| 1,000,000 | Units |
Number of building fires (ignitions) |
N
fi
| 1500 | Fires/year |
Fatalities due to building fires prior to implementation of the safety scheme |
\( N \cdot \lambda_{f,0} \)
| 0.01 | Person/fire |
Injuries due to building fires prior to implementation of the safety scheme |
\( N \cdot \lambda_{i,0} \)
| 0.33 | Injuries/fire |
Cost of injury prevented |
\( \zeta_{i} \)
| 20,000 | £/injury |
Average cost of damage to property per fire |
\( \zeta_{d,0} \)
| 10,000 | £/fire |
Reduction in fatalities due to safety scheme | χf | 90 | %/fire |
Reduction in injuries due to safety scheme | χi | 75 | %/fire |
Reduction in damage due to safety scheme | χd | 90 | %/fire |
Safety scheme upfront cost |
C
0
| 2000 | £ |
Maintenance cost per annum of safety scheme |
m
| 100 | £ |
Discount rate | γ | 3 | % |
Design life of safety scheme |
L
| 50 | year |
Derived quantity | Symbol | Derivation | Metric | Unit |
---|---|---|---|---|
Annual fire occurrence rate |
\( \lambda_{ig} \)
|
\( \frac{{N_{fi} }}{{N_{units} }} \)
| 0.0015 | Fires/building/year |
Fatalities due to building fires post implementation of the safety scheme |
\( N \cdot \lambda_{f,1} \)
|
\( N \cdot \lambda_{f,0} \left( {1 - \chi_{f} } \right) \)
| 0.001 | Person/ignition |
Life preservation benefit |
\( \Delta D_{f} \)
|
\( N \cdot SWTP \cdot \lambda_{ig} \left( {\lambda_{f,0} - \lambda_{f,1} } \right) \)
| 33.75 | £/year/building |
Injuries due to building fires post implementation of the safety scheme |
\( N \cdot \lambda_{i,1} \)
|
\( N \cdot \lambda_{i,0} \left( {1 - \chi_{i} } \right) \)
| 0.0825 | Injuries/ignition |
Injury reduction benefit |
\( \Delta D_{i} \)
|
\( N \cdot \lambda_{ig} (\lambda_{i,0} - \lambda_{i,1} )\zeta_{i} \)
| 7.43 | £/year/building |
Fire induced damage post implementation of the safety scheme |
\( \Delta \zeta_{d,1} \)
|
\( (1 - \chi_{d} )\Delta \zeta_{d,0} \)
| 1000 | £/fire |
Damage reduction benefit |
\( \Delta D_{d} \)
|
\( \lambda_{ig} \left( {\Delta \zeta_{d,0} - \Delta \zeta_{d,1} } \right) \)
| 13.50 | £/year/building |
Benefit arising from safety scheme | ∆D |
\( \Delta D_{f} + \Delta D_{i} + \Delta D_{d} \)
| 54.68 | £/year/building |
Discounted benefit over installation life, L | ∆Dγ |
\( \Delta D_{\gamma } = \frac{\Delta D}{\gamma } \cdot \left( {1 - e^{ - \gamma L} } \right) \)
| 1415.85 | £/building |
Discounted maintenance cost |
m
γ
|
\( {\mathop \sum \limits_{t = 1}^{L} \frac{m}{{\left( {1 + \gamma } \right)^{t} }}} \)
| 2572.98 | £/building |
Costs of safety measure |
C
|
\( c_{0} + m_{\gamma } \)
| 4572.98 | £/building |
J-value for fire safety scheme |
J
T,fi
|
\( \frac{C}{{\Delta D_{\upgamma} }} \)
| 3.23 | (–) |
Derived quantity | Symbol | Derivation | Metric | Unit |
---|---|---|---|---|
Annual fire occurrence rate |
\( \lambda_{ig} \)
|
\( \frac{{N_{fi} }}{{N_{units} }} \)
| 0.0015 | Fires/building/year |
Fatalities due to building fires post implementation of the safety scheme |
\( N \cdot \lambda_{f,1} \)
|
N
\( \cdot 0.5\lambda_{f,0} \)
| 0.005 | Person/ignition |
Life preservation benefit |
\( \Delta D_{f} \)
|
\( N \cdot SWTP \cdot \lambda_{ig} \cdot 0.5\lambda_{f,0} \)
| 18.7 | £/year/building |
Injuries due to building fires post implementation of the safety scheme |
\( N \cdot \lambda_{i,1} \)
|
\( N \cdot 0.5\lambda_{i,0} \)
| 0.167 | Injuries/ignition |
Injury reduction benefit |
\( \Delta D_{i} \)
|
\( N \cdot \lambda_{ig} \cdot 0.5\lambda_{i,0} \cdot \zeta_{i} \)
| 5.01 | £/year/building |
Benefit arising from safety scheme | ∆D |
\( \Delta D_{f} + \Delta D_{i} \)
| 23.71 | £/year/building |
Discounted benefit over installation life, L | ∆Dγ |
\( \Delta D_{\gamma } = \frac{\Delta D}{\gamma } \cdot \left( {1 - e^{ - \gamma L} } \right) \)
| 613.99 | £/building |
Justifiable cost of safety measure |
C
|
\( C{ = }\Delta D_{\varvec{\gamma}} \)
| 613.99 | £/building |