Sensitivity Analysis

Sensitivity analysis is applied to better understand which input factors have the most influence on the model outputs or, in other words, how variations in model outputs can be apportioned to variations in model inputs (Saltelli 2002). Sensi-tivity analysis helps the analyst to establish a ranking of the relative contribution of inputs towards the output results (EPA 2001).

The input factors of the LCC, EI and RA models are analysed by means of a local sensitivity analysis. The principle is depicted in Figure 3.8. Each factor is varied independently by a certain percentage x around its mean value and the relative changes in outputs are analysed. In this way a ranking of the relative contribution of each input factor towards the output results is established. This may then be followed by uncertainty analysis, focussing on the factors with high sensitivity.

Figure 3.8: Principle of sensitivity analysis - adapted from (EPA 2009b)

The advantages of local compared to global sensitivity analysis include its sim-plicity. Since it is straightforward, easy to interpret and does not require high computational requirements, it is the most popular type of sensitivity analysis.

However, it only explores a limited space of input factors around the base case,

and furthermore does not capture interactions between the individual factors. For a more in-depth discussion of local versus global sensitivity analysis see guidance documents (EPA 2001, 2009a) or review papers (Saltelli 2002, Hall et al. 2009, Tian 2013). For an initial screening assessment, a local sensitivity analysis is judged as adequate. Further refinement and global analysis can be done as part of the uncertainty analysis.

3.4.1.1 Sensitivity Analysis of LCC model

The sensitivity analysis of the LCC model is conducted by varying each input parameter f_k one at a time by a percentage x according to Equation 3.11, and determining the changes in the output cost factors Ci.

f_k,x = f_k,0∗ (1 + x) (3.11)

A common measure for sensitivity are partial derivatives, which provide the ratio of the absolute changes in model outputs to the absolute changes in input pa-rameters. The principle of the sensitivity analysis for the LCC model, analysing

∂Ci

∂fk is depicted in Figure 3.9.

Figure 3.9: Sensitivity analysis of LCC model

In the case of a linear additive LCC model, the variation in output forms a straight line with a constant gradient ˜mk. The value of the gradient for each factor f_k can be determined by calculating the partial derivatives analytically or by calculating the difference quotient from the output data:

˜

m_k = ∂C_i

∂f_k (3.12)

= ∆C_i

∆f_k = C_i,x− C_i,0

f_k,x− f_k,0 (3.13)

With f_k,x as the variation of input factor f_k by x % (Equation 3.11) and C_i,x as the result of the cost calculations for f_k,x. Note that f_k,0 and C_i,0 represent the base case, i.e. changing the input factor f_k by x = 0 %.

As mentioned above, the partial derivatives provide a ratio of the absolute changes.

Alternatively, these can be normalised and expressed as relative values (as de-picted above in Figure 3.8), leading to a non-dimensional gradient ¯m. Further-more, it can be beneficial to define a hybrid gradient m that relates the relative changes in input parameters to the absolute changes in model outputs.

The three different types of gradients are depicted in Figure 3.10, with changes in model inputs fk,xon the abscissa and respective changes in model outputs Ci,x

on the ordinate. The advantage of normalising the input values is that this makes them more easily comparable. Whether the model outputs are normalised or not depends on the preferences of the analyst.

Figure 3.10: Comparison of partial derivative, non-dimensional and hybrid gradient for sensitivity analysis

Considering Equation 3.11, Equation 3.13 can be re-written as:

˜

m_k = C_i,x− C_i,0

x ∗ f_k,0 (3.14)

The non-dimensional gradient ¯mk on the other hand is defined as

The non-dimensional gradient ¯m_k therefore relates to the partial derivative ˜m_k by

¯

m_k = ˜m_k∗ f_k,0

C_i,0 (3.17)

Finally, the hybrid gradient m_k is defined as

m_k = ∆C_i

∆x = C_i,x− C_i,0

x (3.18)

The key benefit of using the hybrid gradient mkis that it does not only allow the decision analyst to determine the most relevant input parameter in a straight-forward way (which the non-dimensional gradient ¯mk would do too), but it also allows to identify the absolute change in C_i and not only the relative. Therefore, it is also possible to depict the changes of input factors f_kon different cost factors C_i, as shown in Figure 3.11.

The hybrid gradient relates to the non-dimensional gradient ¯mk through the scal-ing factor C_i,0 and to the partial derivatives ˜m_k by scaling with each base case factor fk,0.

m_k = ¯m_k∗ C_i,0 (3.19)

m_k = ˜m_k∗ f_k,0 (3.20)

Figure 3.11: Hybrid gradient for sensitivity analysis, showing the impact of changes in input factors f_k on different cost factors C_i

The outputs can be analysed individually for initial, operation and maintenance as well as end of life costs. For a linear additive model, the sensitivity of the total LCC for any one factor f_kis the sum of the partial sensitivity values, if expressed by the partial derivatives ˜m_k or the hybrid gradient m_k.

∂LCC

∂f_k =X

i

∂Ci

∂f_k

= ˜m_k,total =X

i

˜ m_k,i

(3.21)

m_k,total =X

i

m_k,i (3.22)

In case of a non-linear model, the local gradient ˜m_k should be calculated through the partial derivative and scaled according to Equations 3.17 and 3.20. For a numerical investigation, the percentage variation should be chosen separately for each input factor, depending on the expected maximum variation. An iterative process between sensitivity and uncertainty analysis may be required.

The results of the sensitivity analysis are used to prioritise the factors for the un-certainty analysis, where the input factors with the highest sensitivity are refined as discussed in Section 3.4.2. Further details are provided in Section 4.4.1.

3.4.1.2 Sensitivity Analysis of Risk end Environmental Impacts

Since the health and safety risks are evaluated on a linear additive basis, the sensitivity of the model is directly related to the value of the individual factors.

Any KPIs with a performance score P_i^j that has an extreme value (high deviation from the baseline) can potentially influence the results significantly. However, for the final score, also the significance scores S_i are relevant, as these are used as multiplication factors. Therefore, the sensitivity of the results towards the indi-vidual KPIs is analysed by calculating the product of the changes in performance score, with respect to the baseline, times the significance score. Consistent with the sensitivity analysis of the LCC model, this product is denoted as m:

m^j_E,i = P_E,i^j − P_E,i⁰
∗ S_E,i⁰ (3.23) m^j_R,i = P_R,i^j − P_R,i⁰
∗ S_R,i⁰ (3.24)

Thus, the sensitivity analysis for the risk and environmental impacts is straight forward and can be done by ranking the respective products according to their values.