Expected cost due to fire

In order to calculate the expected costs due to one fire, one needs to create a model of what can happen if a fire breaks out and how likely each type of event is. The model used in the thesis will be described in this section briefly. For a more comprehensive account, see Johansson 2000a and Johansson 2000b.

The technique used to visualise the model is an event tree technique, meaning that the outcome of a fire is seen as being determined by a set of uncertain events as described in a tree. For example, if a fire occurs in a building it may occur in different areas of the building.

This is modelled by a probability node in the event tree. Figure 21 shows part of an event tree, where an example of the probability node just mentioned is shown.

Fire

The new PK Workshop

The A Workshop

Storage area

ABB Training Center

EMC

The PS Workshop

The P office

The PK Workshop

Remaining parts of the building

Figure 21 Probability node representing the probability of a fire starts in a specific area. The example is taken from the case study performed at ABB.

As can be seen in the figure, a fire that occurs in the building may have started in any one of nine different areas. This is represented by a node and by the events leading from the node. It is important that the probabilities of the events leading from a node sum to 1. Following the probability node concerning where the fire started, come such nodes as those representing the probability that the staff will extinguish the fire, that the sprinkler system will extinguish the fire (if such a system is present), that the fire cells will contain the fire, etc. Since what kinds of probabilities and events are considered in the event tree is dependent on the specific building involved, the modelling of a fire must always be done in an individual way that takes account of this. In the case studies performed at ABB and Avesta Sheffield, the following protection “systems” were considered:

Active systems. Systems designed to actively extinguish a fire should be considered in the analysis. These could be water sprinkler systems, for example, CO2-systems, light water systems, etc. In estimating the conditional probability that an active system will extinguish the fire (i.e. conditional on all preceding events in the event tree), one can use as a point of departure whatever investigations are available concerning the reliability of the system (see National Fire Protection Association, 1976, for example). However, one should remember that the numbers presented in such investigations are estimates of the mean values found for a whole group of systems and that the reliability of a particular system may differ from this.

One should best use the values obtained in investigations of this sort simply as a point of departure in estimating the reliability of a specific system.

Passive systems. System (such as a wall) designed to stop a fire from spreading further in a building but not designed to actively extinguish the fire were likewise considered.

Investigations regarding the reliability of fire-rated walls or fire-rated windows, for example, appears to not be as common as those concerning active systems. This makes it more difficult for the decision maker to estimate the conditional probabilities involved. However, since one

can tolerate probabilities being given in an imprecise way, one can accept the decision maker is representing a conditional probability by an interval or by a probability distribution.

Fire department. Since the fire department can affect the outcome of a fire, it can be represented by the conditional probability that it will succeed in extinguishing a fire. This probability could be estimated in cooperation with representatives of the fire department in question and would probably be estimated in terms of a large probability interval (an imprecise probability). Särdqvist (2000) presents useful information regarding the performance of the fire department in manual fire fighting operations.

Fire type. If a fire starts where there is not much combustible material, it may go out by itself.

This could be modelled as the probability of a fire spreading beyond the initial, limited phase.

Staff. If staff members detect a fire and have the appropriate equipment, they may succeed in extinguishing the fire before it grows to any significant size. The probability of staff members extinguishing a fire would be expected to depend on their training, the amount of fire fighting equipment they have access to, etc.

In looking at different fire protection “systems”, one realises that some of the conditional probabilities involved are very difficult to estimate. This is why one is unable to use a traditional Bayesian approach to estimate these probabilities as precise values. For some of the probabilities, one would simply not be able to settle on any precise value. Instead, one can use the concept of reliability-weighted expected utility (RWEU) explained in the previous chapter. In employing RWEU one estimates a set of plausible values for the different probabilities and consequences and then assigns a second-order probability distribution to this set. Since one assumes that utility and monetary value are the same thing, one has to calculate a set of values for the expected costs due to fire.

The calculation of one value for the expected costs due to fire involves summing the product of the probability and the monetary outcome for each consequence in the tree describing the possible fire scenarios. For example, consider the event tree shown in Figure 22. To calculate the expected consequence (which can be the expected costs if one uses costs as consequences), one first calculates the product of the correct conditional probability and the monetary consequence for each of the consequences shown in Figure 22 and then sums these products.

Figure 22 Illustration of the calculation of the expected consequence from an event tree.

One can note in Figure 22 that if there is uncertainty concerning one or more of the probabilities (or consequences), resulting in a set of plausible values, then the effect will be a set of plausible expected consequences. This is how one deals with the uncertainty concerning the probabilities and consequences; one represents each of them by a set of values that are sufficiently spread to correspond to the uncertainty regarding one’s belief about the parameter involved. In practice this is done by specifying a probability distribution for the parameters and running a Monte Carlo-simulation, resulting in a histogram showing which values for the expected costs due to fire are most probable. As was stated in section 2.3.1 the decision criterion states that the alternative with the highest reliability-weighted expected utility is the best alternative. However, if the decision is not robust (see section 2.3.1) then one should continue one’s analysis in order to obtain more information regarding various of the uncertain parameters.