- Email: [email protected]
A fuzzy decision support system for the economic calculus in radioactive waste management
Information Sciences 142 (2002) 103–116 www.elsevier.com/locate/ins
A fuzzy decision support system for the economic calculus in radioactive waste management Pierre L. Kunsch
a,*
, Philippe Fortemps
b
a
b
ONDRAF-NIRAS, Avenue des Arts 14, 1210 Brussels, Belgium D epartement de Math ematique et Recherche Op erationnelle, Facult e Polytechnique de Mons, Rue de Houdain 9, BE-7000 Mons, Belgium
Abstract A methodology has been elaborated to derive contingency factors for evaluating in a realistic way the costs of first-of-the-kind projects in radioactive waste management (RWM). The paper describes the practical implementation of the fuzzy decision support system (FDSS) and its interface to assist economists in charge of the economic calculus. Uncertainties to be added on top of basic cost evaluations are represented by two contingency factors, respectively, called the P- and the T-factors. The P-factor represents the uncertainties of the project induced by its still incomplete advancement; the T-factor represents the uncertainties caused by the still insufficient technological maturity on which the project is based. Progressive implication rules of the Goguen type are used with the two contingency factors as outputs. Input variables for P and T are given on relative [0,1] scales. Fuzzy logic is also used as front-end for obtaining the two inputs in the course of peer reviews of technology-experts and project-specialists. To that aim, the semantic opinions and past experience of the latter are expressed in the form of conditional and unconditional rules. The credibility of T-experts and P-specialists are taken into account by using the Kleene–Dienes inference. A numerical example on the cost of disposal of high-level radioactive waste in a deep repository is used to illustrate the practical use of the FDSS. This approach is also applicable to other economic assessments inside or outside the nuclear field. Ó 2002 Published by Elsevier Science Inc.
*
Corresponding author. E-mail addresses: [email protected] (P.L. Kunsch), [email protected] (P. Fortemps). 0020-0255/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 2 0 - 0 2 5 5 ( 0 2 ) 0 0 1 6 0 - 3
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Keywords: First-of-the-kind projects; Radioactive waste management; Fuzzy decision support system; Technological maturity; Project advancement
1. Introduction The costs of first-of-the-kind projects are difficult to establish though they are important for defining financing conditions, rates of return, etc. Examples are found in several different fields: information technology, biotechnology, life sciences, spatial projects, nuclear engineering, radioactive waste management [1,2], etc. A common characteristic of such projects is that they have extended time horizons, often resulting from complex hierarchies of sub-projects. Because they are innovative, available experience is limited and there are gaps in technical knowledge and technological maturity. As a result, uncertainties on final designs and costs are very large. In the present paper the authors present a methodology for assessing the uncertainty margins to be taken into account in the economic calculus of such complex first-of-the-kind projects. The main field in which these developments have been made is radioactive waste management (RWM) on which the paper concentrates entirely. A fuzzy decision support system (FDSS) is developed to assist the economic calculus in RWM. A fictive evaluation of a high-level radioactive waste repository sub-project is used as a prototype for testing the model, i.e., the disposal of waste in horizontal underground galleries in a clay host rock at about 250 m below the surface. In Section 2 a fuzzy inference system (FIS) developed in previous conceptual papers is presented. It provides as outputs two contingency factors representing the uncertainties associated with, respectively, the technological maturity and the technical advancement of the considered (sub-)project. The basic cost estimates is multiplied by these two factors in order to obtain a final cost estimate. Section 3 develops the elaboration procedure of the two inputs to the FIS that have to be obtained by organising peer reviews of both experts in the applicable technology and specialists in the considered project. Credibility indices on answers and opinions are taken into account during the elaboration of the two inputs to the FIS. In Section 4, at the occasion of the prototype in RWM, it is shown how a FDSS is assembled in a user-friendly way as an interface between expertsspecialists on the one hand and the economists in charge of the cost evaluations on the other hand. In Section 5 conclusions are presented on the practical use of the FDSS.
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2. The FIS approach of contingency factors in radioactive waste management projects In two previous conceptual papers the authors have introduced the concepts of a FIS to assist the economic calculus in RWM [3,4]. This FIS relies on the recommendations of the Electric Power Research Institute (EPRI) for assessing cost uncertainties of long-term nuclear projects [5,6]. The basic idea is that uncertainties have a dual character. Their roots have to be found in both the incomplete maturity of the basic technology and in the partial advancement of the engineering project itself. Assuming that both technology and advancement are such that the project could be realised ‘‘overnight’’ in t0 ; C0 would give its instantaneous costs, the basic cost estimate of engineers. In reality, however, the project needs to mature during a finite time interval ½t0 ; t , the length of which depends on the actual status of technology and project. To represent the uncertainties associated with this time interval, two ‘‘contingency factors’’ are determined and applied separately to the basic cost estimates. With respect to the two factors, in short project (P-) and technology (T-) factors, the recommendations of the EPRI are as follows: P-factors distinguish four advancement levels (from P1 to P4). The recommended range is given for each level: (P1) (P2) (P3) (P4)
Simplified estimate Preliminary estimate Detailed estimate Finalised estimate
30–50% 15–30% 10–20% 5–10%
T-contingency factors distinguish five maturity levels (from T1 to T5). The recommended range is given for each level according to the availability of past data from similar projects: (T1) (T2) (T3) (T4) (T5)
No or limited data Bench scale data Small pilot plant data Full scale module data Operational data
40% to project specific 30–70% 20–35% 5–20% 0–10%
Experts and specialists are supposed to determine the adequate levels P or T for giving appropriate inputs. The two output values are the contingency factors to be selected in the range Yi and Zi . In this way the final cost estimate is obtained as being given by C ¼ C0 ð1 þ Y Þð1 þ ZÞ:
ð1Þ
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The translation of this approach into a fuzzy inference scheme proves quite straightforward as shown in the previous conceptual papers [3,4]. The EPRI recommendations are translated into two sets of rules as follows: IF P is Pi THEN Y is Yi ;
i ¼ 1; 4;
ð2Þ
IF T is Tj THEN Z is Zj ;
j ¼ 1; 5:
ð3Þ
Fig. 1 shows the rule editor in the fuzzy toolbox of MATLABâ used for the FIS. The second paper [4] has developed this rule approach, assuming that experts can provide the two inputs ðP ; T Þ on a relative [0,1] scale. The main conclusions of this analysis are now reproduced. Taking the example of the P-factor, assume that x is the value for P determined by the peer review of project specialists. R representing a fuzzy implication, the following partial conclusion is obtained with the ith rule: lYi ðyÞ ¼ Rðui ; lYi ðyÞÞ;
ð4Þ
where lY ðyÞ is the membership function of y to Y and ui ¼ lPi ðxÞ. The most adequate rule semantic has to be chosen. Many alternatives do exist in the choice of an appropriate implication for rules [7,8]: certainty rules,
Fig. 1. The set of rules for the T-factor.
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possibility rules, gradual rules, etc. In addition the way of aggregating the rule consequences in the most relevant way has to be examined. To adequately represent the interpretation of experts, gradual rules are preferred to certainty rules. They sound as ‘‘The more x is Ti and the more y is similar to x, the more y is Zi ’’. This corresponds to the natural interpretation of experts ‘‘The less mature the technology behind the project is, the larger the Tfactor must be’’. Because of the smoothness the Goguen implication is preferred to the G€ odel implication as also explained in the paper [4]: G€ odel Rðu; vÞ ¼ 1
if u 6 v;
Rðu; vÞ ¼ v
if u ¼ 0;
Rðu; vÞ ¼ min ð1; v=uÞ
ð5Þ
otherwise;
Goguen Rðu; vÞ ¼ 1
otherwise
ð6Þ
(this is equivalent to Rðu; vÞ ¼ 1 when u 6 v; Rðu; vÞ ¼ v=u when u > v). As a result of the choice of gradual rules, t-norms (min) are to be preferred to aggregate consequences, instead of t-conorms (max) aggregating conjunctive rules, like first attempted [3].
Fig. 2. The Goguen implication on the T-factor.
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Fig. 3. The interval of T-factors (1 þ Z) in function of the input T-level in [0,1].
Fig. 2 shows an example of this implication for determining Z, i.e., the Tfactor, using the five EPRI-rules. Input membership functions (m.f.) are gaussian, while the output m.f. are trapezoidal. This figure illustrates that the Goguen implication provides an output m.f. interpolating between the two most possible Zi’s. This assumes, as discussed in the paper [4], that the overlapping of the m.f. is adequate. a-Cuts are used to obtain contingency factor intervals IðaÞ. The length of the interval represents the doubt at a-level. The resulting contingency T-factor ð1 þ ZÞ is represented in Fig. 3 for a ¼ 0:5. The maturing of the technology brings different positioning in the interval. Conservative positioning when the project is far-away from its realisation corresponds to the upper bound in the interval IðaÞ. Optimistic positioning when the project is moving towards its actual realisation corresponds to the lower bound in the interval IðaÞ. The dynamical properties of the contingency factors correspond in this way to the maturing process of the future project.
3. Organising peer reviews for providing the inputs to the FIS The practical use of the FIS [4] is confronted to the obstacle of providing values for the P- and T-inputs. The question is then to know how to incorporate the opinions and experience of multiple experts and specialists
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participating in the peer reviews. To achieve this result, two steps have to be followed for each T- or P-input. First experts or specialists use proxy measures to indirectly reflect their opinions on the T-maturity or the P-advancement. Second, the latter are inferred from the evaluation of the proxy measures. 3.1. Proxy measures and their evaluations First, proxy measures are defined to represent, respectively, the technological maturity, i.e., the T-level, or the project advancement, i.e., the P-level. Experts in technology and specialists in projects participating in peer reviews have to provide quantitative evaluations regarding these proxy measures. Because the validity of their opinions depends on their personal qualification, it is necessary to combine their evaluations in a way that would reflect their individual credibility. Assuming that opinions on the proxy measure are expressed as unconditional rules such as membership functions, crisp values or distributions of values, a suitable rule for combination has to be defined. Usually unconditional rules are combined using a t-norm, i.e., the AND-operator (min). This idea [9] will be kept here. Weights can be given to the rules [9], but this is not satisfactory, as the result of the combination would then be very dependent on the rather arbitrary credibility indices. A better approach we propose to use is to consider the opinions on the proxy measure as possibility distributions on that variable. These possibility distributions, one per expert or specialist, will act as a restriction on the set of values that are plausible for the proxy measure. The exact value of the latter is obviously unknown; we just have access to the experts’ (specialists’) opinions on this measure. Therefore, on the one hand, their opinions have to be combined by means of a t-norm (e.g. min), since we are looking for values that are compatible with all the experts’ (specialists’) statements. On the other hand, T-experts, or P-specialists, enjoy different credibility levels. This approach can be implemented thanks to the Kleene– Dienes inference defined as follows: Rðu; vÞ ¼ maxð1 u; vÞ;
ð7Þ
where u is the credibility index of the T-expert (or the P-specialist), and v is the membership grade of the proposed proxy measure. In this way, the higher the credibility of the T-expert (or P-specialist) will be, the more certain the expected values and the less possible the unexpected values will be. With this inference, the aggregation of opinions is based on conjunction, e.g. the ANDoperator (min). In this way, the less credible opinions only slightly influence the final results, eliminating excessive sensitivities in the choice of credibility indices.
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3.2. From proxy measures to the T-level (or P-level) Second, the input T-level (or P-level) must be inferred from the corresponding proxy measure by using conditional possibility rules, like: ‘‘The higher is the required T-measure (or P-measure), the higher is the possibility of a limited T-maturity (or P-advancement)’’ Like in Mamdani-control, a conjunction operator (AND-operator) models the implication within each rule. A disjunction operator (OR-operator) performs the aggregation of consequences and combines it with the credibility index. The application of these two inferences is now detailed on the prototype case for determining the two input values to the FIS, i.e., the T-level and the P-level of the high-level waste repository project. To have a robust behaviour, we choose here the min (respectively, the max) for the conjunction (respectively, the disjunction). 3.3. The technology (T)-input to the FIS For bringing the RWM project to a sufficient maturity, money has to be spent for technological research and development (R&D). The corresponding budget is expressed in billions BEF. It should not be confused with the cost of the project itself. The R&D budget is a sensible proxy measure for evaluating the technology (T)-input, i.e., the level of maturity of the technology. Technology experts would provide opinions on the importance of the R&D budget and also say, by the means of conditional rules, how the latter relates to the T-input. Each technology expert would provide his own opinion on the R&D budget in the form of unconditional rules for the specific project. The lower limit of the universe of discourse for these rules would represent the limited level of funding set aside to cover unforeseen expenses at the end of the research programmes. The upper limit would be equal to the total of already spent money and of foreseen future research expenses. These ‘‘unconditional rules’’ describing individual expert advises will often make use of the five grades {Minimal, Reduced, Average, Important, and Maximal} covering the universe of discourse, more rarely crisp values in billion BEF (bBEF). But sometimes experts will be allowed to use their own possibility distributions like ‘‘The R&D budget lies around 4 bBEF’’, etc. Credibility indices of experts in peer review have to be taken into account [9]. When an expert is less reliable, R&D budget values outside his opinion are also possible up to a certain degree. As elaborated earlier in this paper, the Kleene–Dienes implication provides a suitable representation of this situation of
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partial reliabilities [4,8]. For each expert, the result is given by formula (7) in which the input u is the credibility index of the expert, and v the membership grade of his proposed R&D budget. The combination of rules is done with the AND-operator (min). Fig. 4 shows the distribution of budget values for three experts with given credibility indices [0.5, 0.6, 0.7] and specifying distributions for the R&D budget. As required, the least credible expert would have the smallest influence on the final result. The defuzzification uses the maximum mode technique. A crisp value of the R&D budget is finally obtained. Experts then come up with a valuation on how the budget relates to the relative technological maturity, i.e., the T-levels with the five grades T1–T5. This is done in the form of conditional if-possibility rules. Conditional rules map the five grades of the R&D budget onto the T-level universe of discourse {T1–T5}. The implication is modelled as a conjunction AND (min); the aggregation of consequences uses a disjunction OR (max), combined with the credibility in the peer review (min). Fig. 5 shows the resulting inference process to obtain the T-value from the R&D-budget with seven conditional rules from the three experts. The defuzzification uses the ‘‘centre of gravity’’ (COG) approach.
Fig. 4. The aggregation of unconditional rules on the R&D budget for three experts with credibility indices {0.5; 0.6; 0.7}.
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Fig. 5. The inference process to obtain the T-value from the R&D budget. Seven rules are defined by three T-experts.
3.4. The project (P)-input to the FIS The same type of approach is applicable to the determination of the P-input given the necessary adaptations. The manpower effort (expressed in man-years) for bringing the project to its full completion would be here the proxy measure for evaluating the project level (P)-input. Project specialists would be asked to provide opinions on the importance of this manpower effort. Of course, their opinion would in general depend on the T-level representing the technological maturity. The lower limits of the universe of discourse for the specific project would represent the limited human resources set aside to take care for unexpected project tasks just before the project completes. The upper limit would equal the total of already spent resources and of foreseen future manpower to be engaged in the project. Each specialist would provide the requested information either in the form of unconditional rules, should they exclude any influence on their estimate from the T-level, or as conditional rules in the opposite case. The result of the rule would be membership functions using the five grades {Minimal, Reduced, Average, Important, and Maximal}, crisp values, or own
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possibility distributions like ‘‘If the T-level is ‘no or limited data’ then the manpower effort is around 200 man-years’’, etc. As for the R&D budget (Fig. 4), the Kleene–Dienes implication determines the consequence of each unconditional rule for the manpower effort. The respective credibility index is used as input. Rules are combined with the ANDoperator (min). Conditional rules are to be combined with the result using the OR-operator (max) and taking also into account the credibility indices (min-operator). Assume that a crisp manpower effort can be obtained in this way. Specialists then come up with a valuation on how the effort relates to the relative project advancement. This is done in the form of conditional rules mapping the five grades of the manpower effort {Small, Reduced, Average, Important, and Maximal} onto the P-universe of discourse {P1–P4}. As in the case of T-level evaluation, the max-operator is used for combining the rules, considering the credibility indices with a min-operator. The preceding elaboration results in the front-end structure of the FDSS for the case on the high-level waste repository shown in Fig. 6. The P- and T-inputs are then made ready for the Goguen implication shown in Fig. 2.
4. Case study For the sake of testing the FDSS methodology, the project of disposal operations of high-level radioactive waste has been considered throughout the paper. Simulated answers of experts and specialists were generated. Only the details for the resulting T-factor valuation are described in this section. Fig. 7 shows the results from the successive steps. For determining the T-level, three experts were assumed with a credibility vector [0.5; 0.6; 0.7] (top right representation in Fig. 7). Each expert gave one
Fig. 6. The proposed front-end structure of the FDSS for the case on the high-level waste repository providing the T- and P-levels as input values to the FIS.
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Fig. 7. The successive steps in the FDSS for calculating the T-factor.
unconditional rule in the form of a membership function or a distribution for determining the necessary R&D budget (top left representation in Fig. 7). In Fig. 4 and left middle representation in Fig. 7, it is shown that the min-aggregation provided a defuzzified value of a 4.33 billion (bBEF). Of course, this value is only valid for the test purpose. The proxy experts related the budget to the required T-level by formulating seven conditional rules. Combining the rules with the max-operator as shown in Fig. 5 and the down left representation in Fig. 7 provided the T-level 0.27. Fig. 2 and the down right representation in Fig. 7 show how the Goguen implication provided an average value for the T-contingency-factor equal to 1.44 from this T-level input. Combining the T-factor with the P-factor determined in a similar way with P-specialists, a total contingency factor close to 2 was obtained for the disposal project, again as a test value. Note that using the upper bound of the T-factor given in Fig. 3 for the Tlevel equal to 0.27, and the corresponding one for the P-factor, would have produced significantly higher values of the contingency factor. These results look reasonable considering the quite immature status of the disposal technology and project considered.
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5. Conclusions The present paper has demonstrated how the concepts developed in previous papers for the economic calculus of RWM can be made operational in a FDSS. This tool can easily be adapted for the economic calculus of projects with the same characteristics of first-of-kind or prototype, within the nuclear technology, or in other fields of application. Before starting to use the FDSS, a clear picture of the hierarchy of stages and elementary sub-projects being part of the full project must be made clear. This is necessary as elementary stages have in general different time horizons. As an example, the high-level repository project consists of at least four stages, to be decomposed in nine elementary sub-projects on which the full procedure can be applied each time. Suitable interfaces are being developed for collecting information from technology experts (T-factors) and project specialists (P-factor). Proxy mea#132;sures are defined for deriving from peer reviews the inputs to the FIS, i.e., the T-level, or technological maturity, and the P-level, or project advancement. The Kleene–Dienes implication is used to combine unconditional rules with the expert credibility, while the aggregation of opinions is made by conjunction (min-operator). Maximum mode is used for defuzzification. The justification is that the subjective credibility of T-experts or P-specialists must be translated in a smooth way into the possibility grade of their respective opinions. Also space must be left to doubts on these judgements. Conditional rules in assessing proxy measures or deriving the T- and Plevels are combined by conjunction (min-operator), while the aggregation uses the max-operator, combined with the credibility (min-operator). COG is used for defuzzification. The justification is here that, as in control theory, there is a direct translation between the available information (here from experts/specialists) and the kind of action that must be taken (here the choice of a level, given a proxy measure). The inference process in the FIS is rather straightforward using the Goguen implication and the maximum mode defuzzification. The use of gradual rules is here fully justified by the evolutionary character of the project, as its timely path goes from immature plans to full realisation. A further degree of freedom for the dynamic evolution of the assessment is given by the possibility intervals of contingency factors.
References [1] Commission of the European Communities, Costs and Modes of Financing of Geological Disposal of Radioactive Waste, Report EUR-11837, Luxembourg, 1988.
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[2] International Atomic Energy Agency, Assessment and Comparison of Waste Management System Costs for Nuclear and Other Energy Sources, Vienna, Austria, 1994. [3] P.L. Kunsch, A. Ajdler, Determination of funding requirements in radioactive waste management projects using fuzzy reasoning, in: Proceedings of FLINS’98, Antwerp, Belgium, 1998, pp. 353–358. [4] P.L. Kunsch, A. Fiordaliso, Ph. Fortemps, A fuzzy inference system for the economic calculus in radioactive waste management, in: Da Ruan (Ed.), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, A Springer-Verlag Company, Heidelberg, Germany, 1999, pp. 153–171. [5] Electric Power Research Institute, Technical Assessment Guide, EPRI P-4463s-SR vol. 1, Palo Alto, CA, 1986. [6] B. Biewald, S. Bernow, Confronting Uncertainty: Contingency planning for decommissioning, The Energy Journal 12 (1991) 233–245 (special issue). [7] D. Dubois, H. Prade, What are fuzzy rules and how to use them, Fuzzy Sets and Systems 84 (1996) 169–185. [8] A. Fiordaliso, Systemes Flous et Prevision Temporelle, Hermes Science Publications, Paris, France, 1999. [9] E.D. Cox, Fuzzy Logic for Business and Industry, Charles River Media, Rocklands, MA, 1995.
A fuzzy decision support system for the economic calculus in radioactive waste management Pierre L. Kunsch
a,*
, Philippe Fortemps
b
a
b
ONDRAF-NIRAS, Avenue des Arts 14, 1210 Brussels, Belgium D epartement de Math ematique et Recherche Op erationnelle, Facult e Polytechnique de Mons, Rue de Houdain 9, BE-7000 Mons, Belgium
Abstract A methodology has been elaborated to derive contingency factors for evaluating in a realistic way the costs of first-of-the-kind projects in radioactive waste management (RWM). The paper describes the practical implementation of the fuzzy decision support system (FDSS) and its interface to assist economists in charge of the economic calculus. Uncertainties to be added on top of basic cost evaluations are represented by two contingency factors, respectively, called the P- and the T-factors. The P-factor represents the uncertainties of the project induced by its still incomplete advancement; the T-factor represents the uncertainties caused by the still insufficient technological maturity on which the project is based. Progressive implication rules of the Goguen type are used with the two contingency factors as outputs. Input variables for P and T are given on relative [0,1] scales. Fuzzy logic is also used as front-end for obtaining the two inputs in the course of peer reviews of technology-experts and project-specialists. To that aim, the semantic opinions and past experience of the latter are expressed in the form of conditional and unconditional rules. The credibility of T-experts and P-specialists are taken into account by using the Kleene–Dienes inference. A numerical example on the cost of disposal of high-level radioactive waste in a deep repository is used to illustrate the practical use of the FDSS. This approach is also applicable to other economic assessments inside or outside the nuclear field. Ó 2002 Published by Elsevier Science Inc.
*
Corresponding author. E-mail addresses: [email protected] (P.L. Kunsch), [email protected] (P. Fortemps). 0020-0255/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. PII: S 0 0 2 0 - 0 2 5 5 ( 0 2 ) 0 0 1 6 0 - 3
104
P.L. Kunsch, P. Fortemps / Information Sciences 142 (2002) 103–116
Keywords: First-of-the-kind projects; Radioactive waste management; Fuzzy decision support system; Technological maturity; Project advancement
1. Introduction The costs of first-of-the-kind projects are difficult to establish though they are important for defining financing conditions, rates of return, etc. Examples are found in several different fields: information technology, biotechnology, life sciences, spatial projects, nuclear engineering, radioactive waste management [1,2], etc. A common characteristic of such projects is that they have extended time horizons, often resulting from complex hierarchies of sub-projects. Because they are innovative, available experience is limited and there are gaps in technical knowledge and technological maturity. As a result, uncertainties on final designs and costs are very large. In the present paper the authors present a methodology for assessing the uncertainty margins to be taken into account in the economic calculus of such complex first-of-the-kind projects. The main field in which these developments have been made is radioactive waste management (RWM) on which the paper concentrates entirely. A fuzzy decision support system (FDSS) is developed to assist the economic calculus in RWM. A fictive evaluation of a high-level radioactive waste repository sub-project is used as a prototype for testing the model, i.e., the disposal of waste in horizontal underground galleries in a clay host rock at about 250 m below the surface. In Section 2 a fuzzy inference system (FIS) developed in previous conceptual papers is presented. It provides as outputs two contingency factors representing the uncertainties associated with, respectively, the technological maturity and the technical advancement of the considered (sub-)project. The basic cost estimates is multiplied by these two factors in order to obtain a final cost estimate. Section 3 develops the elaboration procedure of the two inputs to the FIS that have to be obtained by organising peer reviews of both experts in the applicable technology and specialists in the considered project. Credibility indices on answers and opinions are taken into account during the elaboration of the two inputs to the FIS. In Section 4, at the occasion of the prototype in RWM, it is shown how a FDSS is assembled in a user-friendly way as an interface between expertsspecialists on the one hand and the economists in charge of the cost evaluations on the other hand. In Section 5 conclusions are presented on the practical use of the FDSS.
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2. The FIS approach of contingency factors in radioactive waste management projects In two previous conceptual papers the authors have introduced the concepts of a FIS to assist the economic calculus in RWM [3,4]. This FIS relies on the recommendations of the Electric Power Research Institute (EPRI) for assessing cost uncertainties of long-term nuclear projects [5,6]. The basic idea is that uncertainties have a dual character. Their roots have to be found in both the incomplete maturity of the basic technology and in the partial advancement of the engineering project itself. Assuming that both technology and advancement are such that the project could be realised ‘‘overnight’’ in t0 ; C0 would give its instantaneous costs, the basic cost estimate of engineers. In reality, however, the project needs to mature during a finite time interval ½t0 ; t , the length of which depends on the actual status of technology and project. To represent the uncertainties associated with this time interval, two ‘‘contingency factors’’ are determined and applied separately to the basic cost estimates. With respect to the two factors, in short project (P-) and technology (T-) factors, the recommendations of the EPRI are as follows: P-factors distinguish four advancement levels (from P1 to P4). The recommended range is given for each level: (P1) (P2) (P3) (P4)
Simplified estimate Preliminary estimate Detailed estimate Finalised estimate
30–50% 15–30% 10–20% 5–10%
T-contingency factors distinguish five maturity levels (from T1 to T5). The recommended range is given for each level according to the availability of past data from similar projects: (T1) (T2) (T3) (T4) (T5)
No or limited data Bench scale data Small pilot plant data Full scale module data Operational data
40% to project specific 30–70% 20–35% 5–20% 0–10%
Experts and specialists are supposed to determine the adequate levels P or T for giving appropriate inputs. The two output values are the contingency factors to be selected in the range Yi and Zi . In this way the final cost estimate is obtained as being given by C ¼ C0 ð1 þ Y Þð1 þ ZÞ:
ð1Þ
106
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The translation of this approach into a fuzzy inference scheme proves quite straightforward as shown in the previous conceptual papers [3,4]. The EPRI recommendations are translated into two sets of rules as follows: IF P is Pi THEN Y is Yi ;
i ¼ 1; 4;
ð2Þ
IF T is Tj THEN Z is Zj ;
j ¼ 1; 5:
ð3Þ
Fig. 1 shows the rule editor in the fuzzy toolbox of MATLABâ used for the FIS. The second paper [4] has developed this rule approach, assuming that experts can provide the two inputs ðP ; T Þ on a relative [0,1] scale. The main conclusions of this analysis are now reproduced. Taking the example of the P-factor, assume that x is the value for P determined by the peer review of project specialists. R representing a fuzzy implication, the following partial conclusion is obtained with the ith rule: lYi ðyÞ ¼ Rðui ; lYi ðyÞÞ;
ð4Þ
where lY ðyÞ is the membership function of y to Y and ui ¼ lPi ðxÞ. The most adequate rule semantic has to be chosen. Many alternatives do exist in the choice of an appropriate implication for rules [7,8]: certainty rules,
Fig. 1. The set of rules for the T-factor.
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possibility rules, gradual rules, etc. In addition the way of aggregating the rule consequences in the most relevant way has to be examined. To adequately represent the interpretation of experts, gradual rules are preferred to certainty rules. They sound as ‘‘The more x is Ti and the more y is similar to x, the more y is Zi ’’. This corresponds to the natural interpretation of experts ‘‘The less mature the technology behind the project is, the larger the Tfactor must be’’. Because of the smoothness the Goguen implication is preferred to the G€ odel implication as also explained in the paper [4]: G€ odel Rðu; vÞ ¼ 1
if u 6 v;
Rðu; vÞ ¼ v
if u ¼ 0;
Rðu; vÞ ¼ min ð1; v=uÞ
ð5Þ
otherwise;
Goguen Rðu; vÞ ¼ 1
otherwise
ð6Þ
(this is equivalent to Rðu; vÞ ¼ 1 when u 6 v; Rðu; vÞ ¼ v=u when u > v). As a result of the choice of gradual rules, t-norms (min) are to be preferred to aggregate consequences, instead of t-conorms (max) aggregating conjunctive rules, like first attempted [3].
Fig. 2. The Goguen implication on the T-factor.
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Fig. 3. The interval of T-factors (1 þ Z) in function of the input T-level in [0,1].
Fig. 2 shows an example of this implication for determining Z, i.e., the Tfactor, using the five EPRI-rules. Input membership functions (m.f.) are gaussian, while the output m.f. are trapezoidal. This figure illustrates that the Goguen implication provides an output m.f. interpolating between the two most possible Zi’s. This assumes, as discussed in the paper [4], that the overlapping of the m.f. is adequate. a-Cuts are used to obtain contingency factor intervals IðaÞ. The length of the interval represents the doubt at a-level. The resulting contingency T-factor ð1 þ ZÞ is represented in Fig. 3 for a ¼ 0:5. The maturing of the technology brings different positioning in the interval. Conservative positioning when the project is far-away from its realisation corresponds to the upper bound in the interval IðaÞ. Optimistic positioning when the project is moving towards its actual realisation corresponds to the lower bound in the interval IðaÞ. The dynamical properties of the contingency factors correspond in this way to the maturing process of the future project.
3. Organising peer reviews for providing the inputs to the FIS The practical use of the FIS [4] is confronted to the obstacle of providing values for the P- and T-inputs. The question is then to know how to incorporate the opinions and experience of multiple experts and specialists
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participating in the peer reviews. To achieve this result, two steps have to be followed for each T- or P-input. First experts or specialists use proxy measures to indirectly reflect their opinions on the T-maturity or the P-advancement. Second, the latter are inferred from the evaluation of the proxy measures. 3.1. Proxy measures and their evaluations First, proxy measures are defined to represent, respectively, the technological maturity, i.e., the T-level, or the project advancement, i.e., the P-level. Experts in technology and specialists in projects participating in peer reviews have to provide quantitative evaluations regarding these proxy measures. Because the validity of their opinions depends on their personal qualification, it is necessary to combine their evaluations in a way that would reflect their individual credibility. Assuming that opinions on the proxy measure are expressed as unconditional rules such as membership functions, crisp values or distributions of values, a suitable rule for combination has to be defined. Usually unconditional rules are combined using a t-norm, i.e., the AND-operator (min). This idea [9] will be kept here. Weights can be given to the rules [9], but this is not satisfactory, as the result of the combination would then be very dependent on the rather arbitrary credibility indices. A better approach we propose to use is to consider the opinions on the proxy measure as possibility distributions on that variable. These possibility distributions, one per expert or specialist, will act as a restriction on the set of values that are plausible for the proxy measure. The exact value of the latter is obviously unknown; we just have access to the experts’ (specialists’) opinions on this measure. Therefore, on the one hand, their opinions have to be combined by means of a t-norm (e.g. min), since we are looking for values that are compatible with all the experts’ (specialists’) statements. On the other hand, T-experts, or P-specialists, enjoy different credibility levels. This approach can be implemented thanks to the Kleene– Dienes inference defined as follows: Rðu; vÞ ¼ maxð1 u; vÞ;
ð7Þ
where u is the credibility index of the T-expert (or the P-specialist), and v is the membership grade of the proposed proxy measure. In this way, the higher the credibility of the T-expert (or P-specialist) will be, the more certain the expected values and the less possible the unexpected values will be. With this inference, the aggregation of opinions is based on conjunction, e.g. the ANDoperator (min). In this way, the less credible opinions only slightly influence the final results, eliminating excessive sensitivities in the choice of credibility indices.
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3.2. From proxy measures to the T-level (or P-level) Second, the input T-level (or P-level) must be inferred from the corresponding proxy measure by using conditional possibility rules, like: ‘‘The higher is the required T-measure (or P-measure), the higher is the possibility of a limited T-maturity (or P-advancement)’’ Like in Mamdani-control, a conjunction operator (AND-operator) models the implication within each rule. A disjunction operator (OR-operator) performs the aggregation of consequences and combines it with the credibility index. The application of these two inferences is now detailed on the prototype case for determining the two input values to the FIS, i.e., the T-level and the P-level of the high-level waste repository project. To have a robust behaviour, we choose here the min (respectively, the max) for the conjunction (respectively, the disjunction). 3.3. The technology (T)-input to the FIS For bringing the RWM project to a sufficient maturity, money has to be spent for technological research and development (R&D). The corresponding budget is expressed in billions BEF. It should not be confused with the cost of the project itself. The R&D budget is a sensible proxy measure for evaluating the technology (T)-input, i.e., the level of maturity of the technology. Technology experts would provide opinions on the importance of the R&D budget and also say, by the means of conditional rules, how the latter relates to the T-input. Each technology expert would provide his own opinion on the R&D budget in the form of unconditional rules for the specific project. The lower limit of the universe of discourse for these rules would represent the limited level of funding set aside to cover unforeseen expenses at the end of the research programmes. The upper limit would be equal to the total of already spent money and of foreseen future research expenses. These ‘‘unconditional rules’’ describing individual expert advises will often make use of the five grades {Minimal, Reduced, Average, Important, and Maximal} covering the universe of discourse, more rarely crisp values in billion BEF (bBEF). But sometimes experts will be allowed to use their own possibility distributions like ‘‘The R&D budget lies around 4 bBEF’’, etc. Credibility indices of experts in peer review have to be taken into account [9]. When an expert is less reliable, R&D budget values outside his opinion are also possible up to a certain degree. As elaborated earlier in this paper, the Kleene–Dienes implication provides a suitable representation of this situation of
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partial reliabilities [4,8]. For each expert, the result is given by formula (7) in which the input u is the credibility index of the expert, and v the membership grade of his proposed R&D budget. The combination of rules is done with the AND-operator (min). Fig. 4 shows the distribution of budget values for three experts with given credibility indices [0.5, 0.6, 0.7] and specifying distributions for the R&D budget. As required, the least credible expert would have the smallest influence on the final result. The defuzzification uses the maximum mode technique. A crisp value of the R&D budget is finally obtained. Experts then come up with a valuation on how the budget relates to the relative technological maturity, i.e., the T-levels with the five grades T1–T5. This is done in the form of conditional if-possibility rules. Conditional rules map the five grades of the R&D budget onto the T-level universe of discourse {T1–T5}. The implication is modelled as a conjunction AND (min); the aggregation of consequences uses a disjunction OR (max), combined with the credibility in the peer review (min). Fig. 5 shows the resulting inference process to obtain the T-value from the R&D-budget with seven conditional rules from the three experts. The defuzzification uses the ‘‘centre of gravity’’ (COG) approach.
Fig. 4. The aggregation of unconditional rules on the R&D budget for three experts with credibility indices {0.5; 0.6; 0.7}.
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Fig. 5. The inference process to obtain the T-value from the R&D budget. Seven rules are defined by three T-experts.
3.4. The project (P)-input to the FIS The same type of approach is applicable to the determination of the P-input given the necessary adaptations. The manpower effort (expressed in man-years) for bringing the project to its full completion would be here the proxy measure for evaluating the project level (P)-input. Project specialists would be asked to provide opinions on the importance of this manpower effort. Of course, their opinion would in general depend on the T-level representing the technological maturity. The lower limits of the universe of discourse for the specific project would represent the limited human resources set aside to take care for unexpected project tasks just before the project completes. The upper limit would equal the total of already spent resources and of foreseen future manpower to be engaged in the project. Each specialist would provide the requested information either in the form of unconditional rules, should they exclude any influence on their estimate from the T-level, or as conditional rules in the opposite case. The result of the rule would be membership functions using the five grades {Minimal, Reduced, Average, Important, and Maximal}, crisp values, or own
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possibility distributions like ‘‘If the T-level is ‘no or limited data’ then the manpower effort is around 200 man-years’’, etc. As for the R&D budget (Fig. 4), the Kleene–Dienes implication determines the consequence of each unconditional rule for the manpower effort. The respective credibility index is used as input. Rules are combined with the ANDoperator (min). Conditional rules are to be combined with the result using the OR-operator (max) and taking also into account the credibility indices (min-operator). Assume that a crisp manpower effort can be obtained in this way. Specialists then come up with a valuation on how the effort relates to the relative project advancement. This is done in the form of conditional rules mapping the five grades of the manpower effort {Small, Reduced, Average, Important, and Maximal} onto the P-universe of discourse {P1–P4}. As in the case of T-level evaluation, the max-operator is used for combining the rules, considering the credibility indices with a min-operator. The preceding elaboration results in the front-end structure of the FDSS for the case on the high-level waste repository shown in Fig. 6. The P- and T-inputs are then made ready for the Goguen implication shown in Fig. 2.
4. Case study For the sake of testing the FDSS methodology, the project of disposal operations of high-level radioactive waste has been considered throughout the paper. Simulated answers of experts and specialists were generated. Only the details for the resulting T-factor valuation are described in this section. Fig. 7 shows the results from the successive steps. For determining the T-level, three experts were assumed with a credibility vector [0.5; 0.6; 0.7] (top right representation in Fig. 7). Each expert gave one
Fig. 6. The proposed front-end structure of the FDSS for the case on the high-level waste repository providing the T- and P-levels as input values to the FIS.
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Fig. 7. The successive steps in the FDSS for calculating the T-factor.
unconditional rule in the form of a membership function or a distribution for determining the necessary R&D budget (top left representation in Fig. 7). In Fig. 4 and left middle representation in Fig. 7, it is shown that the min-aggregation provided a defuzzified value of a 4.33 billion (bBEF). Of course, this value is only valid for the test purpose. The proxy experts related the budget to the required T-level by formulating seven conditional rules. Combining the rules with the max-operator as shown in Fig. 5 and the down left representation in Fig. 7 provided the T-level 0.27. Fig. 2 and the down right representation in Fig. 7 show how the Goguen implication provided an average value for the T-contingency-factor equal to 1.44 from this T-level input. Combining the T-factor with the P-factor determined in a similar way with P-specialists, a total contingency factor close to 2 was obtained for the disposal project, again as a test value. Note that using the upper bound of the T-factor given in Fig. 3 for the Tlevel equal to 0.27, and the corresponding one for the P-factor, would have produced significantly higher values of the contingency factor. These results look reasonable considering the quite immature status of the disposal technology and project considered.
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5. Conclusions The present paper has demonstrated how the concepts developed in previous papers for the economic calculus of RWM can be made operational in a FDSS. This tool can easily be adapted for the economic calculus of projects with the same characteristics of first-of-kind or prototype, within the nuclear technology, or in other fields of application. Before starting to use the FDSS, a clear picture of the hierarchy of stages and elementary sub-projects being part of the full project must be made clear. This is necessary as elementary stages have in general different time horizons. As an example, the high-level repository project consists of at least four stages, to be decomposed in nine elementary sub-projects on which the full procedure can be applied each time. Suitable interfaces are being developed for collecting information from technology experts (T-factors) and project specialists (P-factor). Proxy mea#132;sures are defined for deriving from peer reviews the inputs to the FIS, i.e., the T-level, or technological maturity, and the P-level, or project advancement. The Kleene–Dienes implication is used to combine unconditional rules with the expert credibility, while the aggregation of opinions is made by conjunction (min-operator). Maximum mode is used for defuzzification. The justification is that the subjective credibility of T-experts or P-specialists must be translated in a smooth way into the possibility grade of their respective opinions. Also space must be left to doubts on these judgements. Conditional rules in assessing proxy measures or deriving the T- and Plevels are combined by conjunction (min-operator), while the aggregation uses the max-operator, combined with the credibility (min-operator). COG is used for defuzzification. The justification is here that, as in control theory, there is a direct translation between the available information (here from experts/specialists) and the kind of action that must be taken (here the choice of a level, given a proxy measure). The inference process in the FIS is rather straightforward using the Goguen implication and the maximum mode defuzzification. The use of gradual rules is here fully justified by the evolutionary character of the project, as its timely path goes from immature plans to full realisation. A further degree of freedom for the dynamic evolution of the assessment is given by the possibility intervals of contingency factors.
References [1] Commission of the European Communities, Costs and Modes of Financing of Geological Disposal of Radioactive Waste, Report EUR-11837, Luxembourg, 1988.
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[2] International Atomic Energy Agency, Assessment and Comparison of Waste Management System Costs for Nuclear and Other Energy Sources, Vienna, Austria, 1994. [3] P.L. Kunsch, A. Ajdler, Determination of funding requirements in radioactive waste management projects using fuzzy reasoning, in: Proceedings of FLINS’98, Antwerp, Belgium, 1998, pp. 353–358. [4] P.L. Kunsch, A. Fiordaliso, Ph. Fortemps, A fuzzy inference system for the economic calculus in radioactive waste management, in: Da Ruan (Ed.), Fuzzy Systems and Soft Computing in Nuclear Engineering, Physica-Verlag, A Springer-Verlag Company, Heidelberg, Germany, 1999, pp. 153–171. [5] Electric Power Research Institute, Technical Assessment Guide, EPRI P-4463s-SR vol. 1, Palo Alto, CA, 1986. [6] B. Biewald, S. Bernow, Confronting Uncertainty: Contingency planning for decommissioning, The Energy Journal 12 (1991) 233–245 (special issue). [7] D. Dubois, H. Prade, What are fuzzy rules and how to use them, Fuzzy Sets and Systems 84 (1996) 169–185. [8] A. Fiordaliso, Systemes Flous et Prevision Temporelle, Hermes Science Publications, Paris, France, 1999. [9] E.D. Cox, Fuzzy Logic for Business and Industry, Charles River Media, Rocklands, MA, 1995.