a^da
p\ = Prob(-«> < £i < a\ + y ) = J p£i = erf/ai + y
rn
where erfi is the error function.
(c)Forife=Af
v ' da
Obviously the set of Af outcomes of the project is no longer a continuous Gaussian distribution. It corresponds to a discrete Af variable with given probability distribution, i.e., values of a*, and corresponding probability values pi. These values pi are subject to conditions £ = 1. Otherwise
*=i
they can be arbitrary. Here they have been chosen according to a continuous Gaussian model in order to facilitate application of Bayes' theorem.
In this case the following expressions hold:
M
M
k=l
The objective function OF can now be represented by the following equation (see Appendix B):
AM
(12.3.6)
i=l
Similarly, for the continuous case, the above expression can be modified as follows:
(12.3.7)
N
OF = -5>-l(
t=l
AM
"
}w
The functions a»(u) and pi(u) are defined as:
Oi(u) = Ju p$p\\..p& (12.3.8)
pi(u) = Ju pb°p\\..p& (12.3.9)
JfeeB
where u = u(x) is a function of x, i.e., an application R^-± R x is a vector of dimension M, i.e.,x = (£o,£i 4iV-i).
Each ^i can reach one of the discrete values ak, i.e.,
& = a*< = ]j-jq<fc-l).
where*«' = 1, 2,.-, M
Thenp*' is the probability of occurrence of the outcome £*' at the revision time U and is given by expressions (12.3.2), (12.3.3) and (12.3.4). The indexes i vary between 0 andiV-1.
A and B are the set of indexes ko, ki,..., ki-i defined as follows:
A = lk0 > /l; kl > /l;...; ki-1 <li\ (12.3.10) B = {kO > II; k\ > II;...; ki-l = U) (12.3.11) where hh--.lN define the stopping vector X = (Xi X2 ... XN) in the dis-crete space, i.e.,
. Xi . , mo li = -7-, where da =
TTT-da M-i
The values of li (i=l,2,...^V-l) have to be chosen so that they maximize the objective function OF given by (12.3.7). A computer program has been written to solve this model. A natural optimization procedure in this context is dynamic programming. However, for purposes of illustration, we used the steepest gradient mode.
12.4 Profitability of Swedish R&D Investments in Fusion Energy
We now illustrate the capability of the discrete time model with a case study. The example concerns the profitability of long-run research on energy produced by nuclear fusion in Sweden. This choice is to some
3 It is obvious that the last stopping component value Xs (or IN) has no influence on the objective function OF.
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extent based on a paper by Ståhl (1983), where the profitability of Swedish R&D investments in a fusion energy project was questioned.
Stahl's method consisted of conventional one-shot analysis. The aim — and result — of Stahl's method were to question the expected profitability of a Swedish research program on fusion energy. Our objective is to contrast the result from the model described in the preceding section, using the discrete sequential decision-making process, to those of Ståhl.
We use the same data as Ståhl for our "main" case. We also study the effects of changes in some major parameters in our model, i.e., the number of revision times and the rate of decrease in uncertainty regard-ing research output over time (defined in the model by the exponent a).
Sensitivity analysis of some changes in the main data and their influence on project profitability is also carried out. The data used for the main case, given by Ståhl (1983), and the sensitivity analysis are shown in Table 12.1. Computation of these data and the assumptions introduced in the sensitivity analysis are presented in Appendix C.
Table 12.1 Cases studied
(n denotes the annual increase in the cost of coal; see Appendix C) It should be pointed out that the total cost of electricity produced by fusion energy is comprised of expected production and research costs. In the main case, this total cost is assumed to equal the alternative cost of coal as a mean value. The results of a conventional once-and-for-all decision are given by the value of the objective function, which is SEK 0.403; see Appendix C. This figure refers to the total cost of electricity per kWh (production and research costs for fusion energy at the termina-tion time, 2050). It was already assumed to equal the productermina-tion cost per kWh of coal-fired production. According to this analysis it is not worthwhile to carry out research on the fusion energy project (zero net benefits). This conclusion is in line with the results presented by Ståhl (1983).
However, if revision times are introduced in accordance with the ideas set forth in our model, the results for the main case are shown in Table 12.2.
Table 12.2 Expected costs of electricity production from fusion energy in Sweden. Main case.
Costs SEKI kWh at year 2050.
Shape
The results shown in Table 12.2 indicate the importance of using a sequential decision-making analysis instead of conventional one-shot analysis in funding R&D projects. As for the question of the profitability of Swedish R&D investments in fusion energy, conventional cost-benefit analysis can only provide all-or-nothing answers and in this case, the answer is negative. However, when the sequential model is applied, the answer is postponed until more information is available through re-search. The idea is to invest some resources and search for more knowledge in order to decide whether the project should be stopped or is worthwhile continuing. In this kind of analysis, the way in which uncertainty regarding the outcome of the project changes over time is of particular relevance, i.e., in this model the parameter a.
Thus, for a = 0.1, corresponding to a low rate of decrease with respect to uncertainty, there is no difference between the result obtained using conventional once-and-for-all analysis and the result from our discrete time model. But when these rates are higher, such as when a = 1 or a = 3, the costs of fusion energy will decrease with the number of revision times. This means that when uncertainty regarding the research results decreases quite rapidly, i.e., for values of a greater than 1, it could be worthwhile to continue fusion research. In fact, low research costs through the early revision times provide a considerable amount of information which could be useful in deciding whether the project is to be continued or stopped.
Savings in planned research costs for the fusion energy project can be achieved at the revision time if expectations regarding the project are not favorable. When such a path appears, the project should be stopped and the allotted resources can be saved instead. Such potential savings can be increased slightly by introducing new revision times because the project can be stopped at an earlier point in time when the expected outcome is unfavorable. This fact is reflected in a set of monotonically increasing values of the objective function OF as the number of revision times increases. A limiting version is the continuous revision time model, i.e., revisions are made at every instant during the research period. The results of this model will produce a minimum value for the objective
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function, i.e., an optimistic answer. However, the number of revision times cannot be increased unlimitedly, due to technical reasons as well as inherent costs in the revision procedure (not explicitly taken into account here). This implies that a discrete time model, such as the one used here, seems to be more realistic than the continuous model.
Similar conclusions can be drawn from the sensitivity analysis presented in Table 12.3.
Table 12.3 Expected costs of electricity production from fusion energy in Sweden. Sensitivity analysis. Costs (SEK/kWh at the year the research project is completed
N = number of revision 0
Column N=Q (no revision times) represents the alternative production cost of coal at the time the research is terminated. This column indicates that
1. if the research period is reduced, the production cost will also decrease;
2. if the interest rate is increased, obviously the production cost will also increase;
3. an increase (decrease) in the costs of coal energy naturally leads to an increase (decrease) in alternative production costs.
It was assumed in all cases that, at the initial point in time, the cost of producing electricity from fusion plus research costs will be equal to the alternative production cost at the end of the research period. As in the main case, when revision times are introduced in the economic evalua-tion, savings can be achieved in the expected value of the production