3.6 Conclusions of Chapter 3
4.4.1 Sensitivity Analysis of LCC Model
The sensitivity analysis for the LCC model was set up to investigate which factors contribute most to the variability of each model, namely the baseline model (con-ventional steel) as well as the two new material options (HSLA and composites).
This was approached with the methodology outlined in Section 3.4.1.1.
In order to investigate those aspects numerically, the input factors were varied one at a time by a certain percentage x and the impact on the LCC results was analysed. Since the LCC model is linear, the variation of an input factor fk by any given percentage leads to a linear variation of the results, with a constant gradient mk. Therefore, the percentage variation considered does not influence the result. For demonstration purpose a variation of ±200 % was chosen.
As discussed in Section 3.4.1.1, the hybrid gradient mk was seen as a suitable measure for the sensitivity, because it relates the relative percentage changes of the input factors to the absolute changes in the model outputs. mk was calcu-lated according to Equation 3.18. Subsequently, the input factors were ranked according to their sensitivity on the results.
The sensitivity analysis was not only undertaken numerically, by varying the input factors one at a time and determining the variability of the outputs, but also analytically by deriving the partial derivatives of the LCC expression according to Equation 3.12. The full results are provided in Appendix B.2, as well as the Visual Basic for Applications (VBA) scripts that were written to support the analysis (Appendix B.4).
Sensitivity of Baseline Model The aim of the sensitivity analysis for the baseline model was to identify the contribution of each factor towards the LCC for conventional steel. As indicated in Table 4.12, the LCC model for conventional steel only depends on baseline factors φk, hence only these were investigated. The results are provided in Table 4.29, sorted according to the order of magnitude of the sensitivity (calculated with the hybrid gradient mk, Equation 3.18).
The sensitivity of the baseline model (in other words the contribution of each factor towards the LCC results) could alternatively be determined in a simpler way: Since some of the input factors (namely φ1, φ6, φ8, φ10 and φ11) were originally defined as ratios per LCC, their input values already provided the sensitivity, as becomes apparent when comparing the values in Table 4.13 with the results of the sensitivity analysis given in Table 4.29. The ratios of other input factors (namely φ2, φ3 and φ5, which were originally defined per initial costs) had to be scaled to the common denominator LCC by multiplication with φ1. This
Table 4.29: Sensitivity of input factors for LCC baseline model
Rank Factor Gradient m
1 φ6 BL fuel costs 6.71 × 10−1
2 φ1 BL initial costs 1.76 × 10−1
3 φ8 BL maintenance costs 1.53 × 10−1 4 φ3 BL material costs 1.32 × 10−1 5 φ5 BL manufacturing costs 3.70 × 10−2
6 φ2 BL design costs 7.04 × 10−3
7 φ11 BL revenues −8.41 × 10−4
8 φ10 BL disposal costs 1.40 × 10−4
determined the contribution of these factors towards the LCC results, which was again equal to the sensitivity, calculated with the hybrid gradient mk.
The fuel costs were found to have the highest sensitivity with regard to the base-line LCC, followed by the initial costs and the maintenance costs. The factors were compared by their order of magnitude: The highest contributing factor φ6
(fuel costs), was found to be 3 to 5 times more important than the factors φ1 (initial costs), φ8 (maintenance costs) and φ3 (material costs). Factor φ5 (man-ufacturing costs) was almost 20 times less important than φ6 and all subsequent factors were found to be 2 to 4 orders of magnitude less important than the most important factor φ6. Only the factor φ11 (baseline revenues) had a minus sign, which meant that, in contrast to all other factors, increasing this factor would result in a decrease of LCC. A couple of baseline factors did not contribute to the baseline LCC model, namely φ4, φ7 and φ9. However, since these were de-rived from the same sources as the other baseline factors, they were nevertheless notated in the same way.
Sensitivity of HSLA and Composite Materials As indicated in Table 4.12, the LCC models for HSLA and composites were dependent on all types of input factors fk: The focus of the analysis was set on the limiting factors λk and the change factors δk, as these were identified as the factors that might vary. For example, the amount of baseline material that can be replaced by the new material options might increase in the future, or the different cost factors might change, depending on economic development and market forces. The baseline factors φk
on the other hand were used to relate λk and δk to the LCC. The factors φ4, φ7
and φ9 that had originally been notated as baseline factors, because they were derived from the same source as the baseline model, were included as well. The results are presented in Tables 4.30 and 4.31.
Table 4.30: Sensitivity of input factors on LCC for HSLA
Rank Factor Gradient m
Table 4.31: Sensitivity of input factors on LCC for Composites
Rank Factor Gradient m
11 δ19C Change in revenues 0.00
Another option is to visualise the gradients as depicted in Figures 4.16 and 4.17.
The variation of LCC is plotted as a function of the percentage variation x of the most contributing input factor fk (gradient mk > 1 × 10−3, as indicated by the dashed lines in Tables 4.30 and 4.31). In both figures the baseline factors φk are depicted with dash-dot lines, whilst limiting factors λk are depicted with dashed lines and change factors δk are depicted with solid lines.
-200 -150 -100 -50 0 50 100 150 200 92
96.21 100
LCCHSLA(%)
Variation x (%)
λH13 δH17 δH16 δH14 δH15 λH12 φ7
Figure 4.16: Sensitivity of input factors on LCC for HSLA (LCCH)
Again the input factors were compared with regard to their importance, identified by the order of magnitude of the sensitivity. The three most important factors in both cases were the two limiting factors λ13 (weight ratio) and λ12 (max conv steel replaced) and the baseline factor φ7 (cost per weight savings), even though their ranking order varied between HSLA and composite materials. Any of these three factors was found to be at least three times more important than all the other input factors. However, if they were compared to the sensitivity gradients of the baseline model, it could be observed that the four highest ranking baseline factors were all at least 1.5 times more important than the highest ranking limiting or change factor. The sensitivity gradients of the baseline factors for the LCC models for HSLA and composites were found to be very similar to the ones for conventional steel, with the full results provided in Appendix B.2.
Accordingly, it was concluded that the validity of the baseline model needed to be investigated further. For all three LCC models, the same four input factors were identified as contributing most to the sensitivity of the LCC model:
1. φ6: Baseline fuel costs,
-200 -150 -100 -50 0 50 100 150 200 90
91.55 93
LCCComposites(%)
Variation x (%)
λC13 δC17 δC16 δC14 δC15 φ4 φ9 λH12 φ7
Figure 4.17: Sensitivity of input factors on LCC for Composites (LCCC)
2. φ1: Baseline initial costs,
3. φ8: Baseline maintenance costs, 4. φ3: Baseline material costs.
One factor that was not accounted for in the sensitivity analysis, is the assumed lifetime of the ship structure. This is due to the fact that it is not an explicit factor in the model, but incorporated in the baseline LCC model. However, it is one of the key assumptions in the original model and thus has a large effect on the results. Therefore, it should be investigated separately as to how much it contributes to the high sensitivity associated with the baseline model factors.
This is discussed qualitatively in Chapter 5.
One more assumption that was incorporated in another factor is the fuel price, which is contained within the baseline split of costs. The fuel price is a highly volatile value and impacts significantly on the operational costs, and accordingly on the total LCC. Thus, a discussion about the fuel price is also included in Chapter 5.
In the next step, the uncertainty associated with the most sensitive factors was investigated, in order to gain more clarity about which factors are, or might be a key issue (refer to Figure 3.7 in Section 3.4).