Table 4.3: Comparison of optional tunnelling technologies.
Option A B
Expected delay [months] 0.125 0.25
Risk of Investor Contractor Investor Contractor
(utility) (utility)
Risk 12 500 € 19 000 € 79 550 25 000 € 38 000 € 159 270 Cost saving against op. A 0 0 0 40 000 € 60 000 € 60 000* Increase of risk against op. A 0 0 0 12 500 € 19 000 € 79 720 Cost saving – increase of risk 0 0 0 27 500 € 41 000 € -19 720 *the utility of cost saving is considered to be linear, i.e. .
The options are compared based on the difference of risk and cost savings. If for option B the cost saving is higher than the increase of risk with respect to option A, the option B is more advantageous. On the contrary, if for option B the cost saving is lower than the increase of risk, it is better to select option A.
As is evident from the results, the decision is disputable. If the utility is considered to be linear to the potential losses (as in the case of and ), the option B appears to be more advantageous for both the investor and contractor. However, taking into account the contractor’s aversion to higher losses, which is modelled by the power utility function (and included in ), the option B turns to be risky for the contractor. In this case, the interests of the investor and contractor are contradicting and the contractor is likely not to accept the alternative technology (option B).
4.3 Summary and discussion
A model for probabilistic estimate of damages caused by tunnel construction failures is proposed. It is applied for estimating the delay due to failures. The model takes into account the variable failure rate in different sections of the tunnel. Additionally, the epistemic uncertainty in estimation of the failure rate is included in the model by introducing the variable human factor. The variable represents the overall quality of the planning, design and construction, which affect the probability of failure occurrence. The influence of these factors is uncertain in the design phase giving rise to the uncertainty in the selection of the failures rates. The model is applied to a case study of tunnel TUN3 (see Section 4.2). The final estimate of the delay due to failures for this tunnel is shown in Figure 4.7.
The estimated delay can be used for quantification of risk, as illustrated in Section 4.2.3. The risk is quantified for three cases. In case 1 and 2, the risk is defined as the expected financial loss (from the point of view of the investor and contractor, respectively). The loss is expressed as a function of delay. For such analysis, only the estimate of expected value of the delay is needed. In case 3, the risk is defined as negative expected utility. By introducing the utility the aversion of the contractor to high financial losses can be taken into account. For modelling the utility, the full probabilistic estimate of the delay is used.
Finally, an example decision-making process is presented in Section 4.2.4. Two alternative tunnelling technologies are evaluated based on comparison of their risk and costs. It is shown that the decision would differ for the two alternative definitions of risk (cases 2 and 3). For the case 3, the risk aversion of the contractor outweighs the benefits from the cost savings.
The example of risk estimate and decision-making presented in this chapter aims at demonstrating how the probabilistic estimates of construction time/cost should be utilized in the tunnel project management. The results of the models presented later in this thesis might be utilized in the same way. The selected monetary values as well as the utility function are purely illustrative. In reality, other factors should be included in the decision making-process, the modelling of construction costs should be improved and factors such as environmental or social impacts should be included. A detailed investigation into the decision-making concepts is, however, beyond the scope of this thesis.
A complex model for modelling of the tunnel construction time is presented in this section. The main requirements on the model are the following: (1) It should consider both types of uncertainties, i.e. the usual variability of the construction process and the extraordinary events – see Section 2.7. (2) The model should consider the common factors that systematically influence the construction process, such as human and organizational factors. These factors introduce stochastic dependence into the performance at different phases of the construction. The significant influence of such dependences on construction performance estimates is shown for example in van Dorp (2005), Yang (2007) and Moret and Einstein (2011). (3) The model should allow for making full use of data available from previous projects, such as advance rates and costs recorded during excavation of tunnels under similar conditions. In this way, the know-how can be systematically managed. (4) The methodology should facilitate the easy updating of predictions when new information on the analysed project (e.g. geotechnical investigations, advance rates and costs observed after commencement of excavation) is available. (5) The model assumptions and involved simplifications must be properly understood and described. This is important in probabilistic modelling where results are difficult to validate by experiments and must therefore be well reasoned.
Many of the requirements could be satisfied by means of the commonly used MC simulation based or analytical approaches. One can model the occurrence of both types of uncertainties (requirement 1), it is also possible to include the dependences introduced by common factors (requirement 2). The DAT model (Einstein, 1996) or the models by Isaksson and Stille, 2005) and by Steiger (2009) can fulfil these requirements – see Section 3.2. Updating of the predictions (requirement 4) based on observed performance during the tunnel construction has been also presented in the literature, usually by means of Bayesian analysis (Chung et al., 2006).