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INTRODUCTION

say , if the value of a basic variable in question is l ess than the break- even value, which is predetermined from the regulatory value, then an incentive is paid to the fa cility, otherwise a penalty is paid by the facility. In most cases , this sort of in spection costs a great deal since i t takes a lot of time and may r e duce an availability factor of the facility. Th e penalty i nc ludes the loss due to this, and, on the other hand , the amount of in~entive should b e hi gh e nough to e ncour age facility effo rts t owa rds the principle of ALAP. It implies th a t both th e a mounts of inc e ntive and p e n a lty a r e fairly hi gh, and th e r e fore an obs e r vation plan for th e dec ision- making must b e designed carefully .

Th e p robl e m of assessing the risk/benefit associated with nuclea r technology cannot be limite d to a technical app r oach , but rath er it involves deeply aspects conce rning human attitudes, value scales , style of life and so forth. Th e re i s , then, the need to treat the problem with an integrated approach, and ther eby to find conc rete measures encouraging a shift in public opinion and the industrial efforts towards a greater awareness and sensitiveness to the problem . This paper is to p ropos e a mathematical model as one of those approaches . While a variety of social impacts are involved in nuclear technology, the inves tigation is focused on environmental impacts. Routine and accidental rel eases of radioactivities and hazards from plu tonium dispersal are representatives of env ironmen tal impacts . Technically, these impac ts are often discussed in terms of the magnitude of risk , i.e. the product of p robab ility of an e vent and its consequence , and a coup l e of f eatures of the risk are worth noticing; fir s t, in assessing t h e p r obabil ity , one has to take into account very low- probabil isti c and sometimes even hypothetical situat ions, and second, long - term int eg rati ons of sc ientific knowledges are r equired for thoroughly eva l uating the consequences. From these features , though regulatory values are set for each factors r es ulting in a risk, the even lower magnitude of ri sk is r ecommended (the p rincipl e of as low as practicable ).

Since a basic variable concerning any environmen tal impact behaves stochastically , the choice of the act must be made under uncertainty. One determines a theoretically optimal observation plan for given amounts of incentive and pena lty , by maximizing th e net gain of observation, defined by the Bayesian decision theory. The value of in centive or penalty for a unit of basic variable is repr esente d in monetary terms and can be r egarded as a sort of social cost associated with the correspondi ng impact . Th e mode l assesses th e social cost by find ing the indifference between a theoretically optima l plan and the actual or p ractical plan. MATHEMATICAL MODEL Two-Ac t Problem

The model presented here is intended to assess the impact s not in such a direct way but in an indirect mann e r. It is supposed that , once the values of regulation criteria are set, a regu l ation authority must observe measurements in each nuclear fa c ilities so that it can verify the validity of the meas u rements . The verification procedure is formulated as a two - act decision problem . I . e., as a result of observation, a regula tor ' s act is chosen to acc ept or reject the facility measurements, a nd, in case th a t act of ' rejection is adopted, some furth e r action , an ad hoc inspection wil l be taken in order to arrive at a conclusion; the con clusion is describ ed here as ei ther payment o f incentive or l evy of penalty . That is to

Let sand sb denote a basic variable in question and the maximum permissible value for it, respective ly , and suppose that a regulation authority is to make a decision on the following two-act problem ; Act- l: Reject the facilit i e ' s measurements and do an ad hoc inspec tion. As a result of the inspection , if S < sb , then pay an incentive KI (s) to a facility , and if s > sb , then levy a pena l ty Kp(s) on a facility , and Act-2: Accept the facilitie's measuremen ts and do

841

842

A. Suzuki and R. Kiyose

nothing further. This implies that the costs of acts, and C2 are represented by,

KI(O

~

< ~b

-Kp(~)

~

> ~b'

must be satisfied. The difference, E(1)VOI is defined as the value of observed information.

C1

C1(~)={

(1)

and (2)

Decision under Uncertainty In view of the stochastic behavior of the basic variabl e , the decision must be made under uncertainty. To formulate it, the Bayesian decision theory is applied. Specifically, let PtO) (~) denote the prior distribution of ~ , and an optimal act under P tO) (~) is obtained from the criterion:

min [E(O)Ci(~) i={1,2}

1,;, Ci ( ~ ) ,p(O)

( ~ ) d~ l.

=

P (0) (~) • P (x I ~ )

/~: P(O) (~) .P(xl~) d~ ,

(la)

COO(1)

is the cost of observation plan.

Assessment of Social Cost Maximization of Eg. (la) yields a theoretically optimal plan, resting on th e amounts of incentive and penalty. These pa rameters should be eva lu ated not on l y from the technical point of view but rath er fr om the f eas ibility in practice. The refor e the model assesses them by finding t he relation:

(11)

(4)

and it is called a posterior distribut ion . An optimal act under P (1) ( ~ ) is obtained simply in the same manner as Eq. (3). Value of Information Obviously the distribution P (1 ) ( ~ ) is more reliable than P tO) ( ~ ). This fact can be formulated as follows. Let the loss due to act-i, Li be defined by the difference between the cost of act-i, Ci and the cost of the optimal act under the perfect information, C*. i. e. ,

i

Needless to say, the value of observed information depends on the way to observe measurements: accuracy of a measuring device, number of measur e d data and so forth. To make higher the value of observed information, one needs to improve the way and it requir es mo re costs for observation. It follows from this t hat an optimal observation plan can be determined by maximizing the net gain o f observation NCO(1) , defined as,

where

D)

Since in most cases the prior distribution is assigned tentatively, it is not enough reliable to make the decision. By observing the conditional probability p (xl ~) from actual measurements x , the distribution of ~ , P (1) ( ~ ) is recalculated by the formula:

P (1) (~)

Optimization of Observation Plan

1,2.

(5)

where, TOP(1) is the theoretically optimal observation plan, esti mated by the above procedure , and POP(1 ) is the plan which exists actually or is cons ide red to be practical. A set of the values on KI ' Kp and ~b , being subject to Eg. (11), is an expression of the socia l cost. ILLUSTRATED EXAMPLE*

Routine In spection Obse rvation of n sam les from l\ct of I iAcceptance_i

For our problem,

L1

(~)

=

KI (E, )

E; < ~b

0

E; > ~b '

0

~

(6)

{

and

L2 ( ~ )

I I,d

< ~b

(7) Kp (~)

hoc Inspection I iSome Further Actio~ -I-

~ > ~k

The optimality condition (3)

~~

is equivalent

yes -l-

to

E(j) VPI

min [ E(j)Li(~) i={1,2} t;, Li(E;),p(j) (E;) dt;l ; j=O , 1,

fPayr,1ent of jIncentive '

/Levy of Penaltyj

(B)

and E(j)VPI is called the cost of uncertainty or the value of perfect iLformation under the distribution p(j) (~). To make sense the observation plan to reassess the distribution of ~, the inequality: (9)

Figur e 1:

I ns p ection Pro cedu re base6 on Problem in In centive Penalty system , E(o): Exo ectat i on of sta ti cs Ob Br~ak-even value of O. ~wo-~ct

* For further information, see Ref.

(1).

cS

843

A model for assessing social imp acts

Assumptions For illustration , l et th e possibility of nuclear mate rials diversion to non-peace ful purposes be taken as one of t he e xamp l es of social impacts. To be against nuclear materials diversion, a n u clear material safeguards sys t em must b e en fo ~ced so that it can verify whethe r or not the material unaccounted for in a nuc l ear fu.e l p r ocess ing facility i s reasonable. Acco rding to t he two-act problem ~ entioned above , the verification p rocedure can be desc ribed schematically as in Fig. 1. For s implification , the foll~~ing assumptions are made; (1) The basic variable S is the diffe r ence (0) between the materials ~n accoun t ed for estimated from inspector' s measurements and from facilitie's measurements. (2) Errors associat ed with the rneasur ements are divided into two components: systemat i c and random, the statist ical behaviors of which are a lr E,ady -known . (3) The prior d istribution of C can be ass ign ed to be a Normal distribution 2 ( E(O ) ( 0 ), 0 (0 ) ( 0 )), and the observed information p (xlo ) is Norma l. (4) Only a samp l e size of i nspecto r' s measu r emen ts, n for the popu l at ion size N i s an optimized parameter on t:'e observation p l an. (5) The b r eak- even value 0b is f ixed in such a way that the minimum critica l mass for a nuclear weapon is defined as the threshold amount pe r year , a ha l f of which i s supposed to be the maximum permis sible value for a year on the mat,erials unaccounted for in a facility, and the max imum pe rmi ssible va l ue averaged per piece is the break- even value. (6) The incentive and penalty are r ep r esented in the simp l est form:

It,X

6X 0

2

6y

(0) ( 0 ) ,

1 /

02 (t,X ) ,

= 0 2 (68 )

(6X)

68

0

(6xl + . .. + 6x n ) / n ,

=

=

2

1 /

+ 0 2 (6y ) ,

th e difference in the mean of random e rr o r s be t wee n inspector's and facility mea surements, and the difference in the systematic errors between inspector's and facility measureme nts.

1 (0 ), ItJi and 1 (1) are cal l ed th e info rmati on quantit i es prior to, from, and poste rior t o obse r vat i on . Furthermore, the distribution o f E (l ) (0 ) is the Normal one with the expecta ti on E ( E(1 ) ( 0 )) and the variance 0 2 (E (ll ( 0 )). (16)

E (0 ) ( 0 ) ,

and 02 (E(] )( 0))

0

2

E68 = 0

2

(0 )( 0 )

(1 + E68 +

/ where , Ey

= 0

(y)/ 2

0

2

n1 Ey

(0) (0 ),

(68) / 0

2

(0 )

N- n N - 1 ),

(17)

and

(0 ) .

Given the distribution of the posterior mean in Eqs.(16) and (17), the formula below is used to compute the e xpected value o f obse rv ed information defined by Eq. (9) :

wh e re,

KI (O)

=

Kp (o )

DE(1 )

kR = constant.

(1 2)

C (DE (l )) is the loss integ ral function (Ref. 2) •

And add it ionally, (7) The cost of observation is by,

Ks , ks

=

constant.

~pp roximated

Then the net gain of observation NCO (1) in Eq.(lO) is easily g i ven from Eqs.(13) and (1 8 ). Th e value of n which maximizes NCO (l) is a theoretically optimal samp l e s iz e and hence y i e ld s a theoretically optimal observation plan .

(13) Computational Results

Under th ese assumptions, the pos terior distribution of 0 afte r observing n samp l es (6xl , "', ~~n ) i s described as the mean E(1 )(O) and the variance 02 (1) (( ), given as f o ll ows ; I (O)E (0 ) ( 6 ) + 16;; U:.x - E (68 ))

1 (0 ) + I tJi (14) and 0

2

(1) ( 0) =

where,

1/1 (1)

(1 5 )

Numerica l examp l es are shown in Table 1 to demonstr ate the computationa l results f o r some kinds o f nuclear fue l p r ocess ing facilities. Th e theoretically opt imal sample s ize for the nuclear fu e l fabrication facility is 205 fuel pins pe r campaign and 5% of the population size. For the fue l reprocessing fa c ility, t he optimal solution is 42 fuel assemblies pe r campaign and more than 40% of the pop ulation size. In case of the fu e l enri chment fa c ility , the figur e correspo nds to 9% of the popula tion size.

A. Suzuki and R. Kiyose

844

Table 1 provides not only the optimal sample sizes but also the costs of uncertainty prior and posterior to observation, indicating that the costs of remaining uncertainty are much higher for the reprocessing and enrichment facilities than for the fabrication facility. This results mainly from the fact that the systematic error of measurements is significantly involved with the former two while it is nil for the latter. An optimal solution is obtained from numerical calculation. Taking into consideration the range of values on relevant data however, an asymptotic solution, n(a) is to be used virtually in place of a numerical

Table

1:

solution, and it is written as;

G(O) n(a)

[--

2

KR

N

_·0(O)(0)·t:y-- l

ks

l/2

N-l 1 3/

4 ,

(19)

N-l

where it is assumed that E(O) (0) = ob. Bquation (19) is an expression of the relationship between an optimal observation plan represented by n(a) and the social cost in terms of kR. If the theoretically optimal value on n obtained here can be considered incidentally to be equal to the actual or practical sample size, then the values on kR in Table 1 give the corresponding social costs.

Illustrated Examples on Observation plan for Nuclear Materials Safeguards

Uranium-Fuel Fabrication Facili ty

Fuel Reprocessing Facility

Fuel Enrichment Facility

Specification: Capacity/year 100 tons of U02 No. of Campaigns 5/year 60 days Campaign Interval Total Pieces/Campaign 4165 fuel pins Population Mean 4482.9g of 2.5w/oEU Expectation of random error Og Standard Deviation of random error 17.8g Expectation of systematic error Og Standard Deviation of systematic error Og

200 tons of U 5/year 60days 100 fuel assms. 3000g of Pu Og 30g 30g 42g

1972 t of UF 6 l/year 300days 986 bombs 2000kg of UF6(4%) Okg 4.0kg Okg 3.6kg

Inspection Parameters: Threshold Amount/year Incentive, Kr(O) Penalty, Kp(OTA) Sampling Cost/Piece, ks Threshold Amount/Piece, 0TA Break Even Value, ob Social Cost/Piece, kR

25 kg of 23 Su $ 4XIO S $ 4XIO S

$ 4 50g of 2.5w/oEU 25g of 2.5 w/oEU $ 1.6 x I0 4 /g

8 kg of Pu $ 4xIO s $ 4xIO s $ 40 16 g of Pu 8 g of Pu $ 5 x IO'/S

25 kg of 23S U $ 4xIO s $ 4xIO s $ 40 938g UF6(4%) 469g UF 6 (4%) $ 8.53 x I0 2 /g

Computational Results: Prior Distribution, PeO) (0) Expectation Standard Deviation Optimal Act under p(O)( O) Cost of Uncertainty under p(O) (0) Optimal Observation Plan Optimal Sample Size Net Gain of Observation Observation Results Mean Standard Deviation

25g of 2.5w/oEU 5.63g of 2.5w/oEU Either $ 3.58xI0 4 205 fuel pins $ 3.43 x I0 4 24g of 2.5w/oEU 1.36g of 2.5w/oEU

Posterior Distribution, PI]) (0) Information Quantity prior to Obsvtn. Information Quantity from obsvtn. Information Quantity posterior to obsvtn. Optimal Act under P(l) (0) Cost of Uncertainty under P(l) (0)

3.23xIO- 2 5.40xIO- 1 5.72xIO- 1 Accept $ 2. 77xI0 3

8 g of Pu 100 g of Pu Either $ 1. 99xI0 6

42 fuel assms. $ 1.84xI0 6 37 g of Pu 42.2g of Pu 1. 00xlO- 4 5.64xIO- 4 6.64xIO- 4 Accept $ 7.53 x IO S

469g of UF6(4%) 5000q of UF 6 (4%) Either $ 1.70xI0 6 85 fuel bombs $ 1. 37xI0 6 450g of UF 6 (4%) 3624g of UF6(4%) 4.00XIO- 8 2.60xIO- 8 1.16xIO- 7 Accept $ 9.94xIO s

A model for assessing social impacts CONCLUDING REMARKS An actual observation plan for nuclear materials safeguards is designed from considering more sophisticated situations: multi-attribute samples, multi-mode operational schemes, inter-facility relationships and so forth. Therefore one cannot straightforwardly apply the results here to actual situations. However one can get the feelings about the special features of nuclear materials accounting system as compared to the non-nuclear cases, where the value of kR is fixed simply at the level in commercial market. Since the market price of plutonium, for instance, is in the order of $lO/g of Pu, the result for the reprocessing facility indicates that, for the purpose of nuclear materials safeguards, one needs to put the value of plutonium hypothetically thousands times as much as the market price. This paper gives the theoretical framework of assessing social or environmental impacts of nuclear technology, the summary of which is as follows; (1) a two-act problem concerning the incentive-penalty system is supposed to formulate the principle of ALAP, (2) an observation plan to make decision on the problem is optimized by making use of the Bayseian decision theory, (3) the optimized solution resting on the amount of incentive or penalty is compared with an actual or

845

practical plan, and then, (4) by finding the indifference between the two plans an impact is assessed in monetary terms. As regards the 3-rd step of this procedure, however, the model does not provide the details since it is beyond the scope of this paper. If there exists an actual plan, one can easily compare it with the results from this theory. If there does not or in the process of making it, the feasibility of it in practice must be studied by another model or by different approaches. ACKNOWLEDGEMENT The authors would like to express their gratitude to Professors Y. Yamamoto and K. Oshima, and Dr. R. Imai for their helpful discussion and encouragement. REFERENCES (1) A. Suzuki, An Incentive-Tax Model for Optimization of an Inspection Plan for Huclear Materials Safeguards, IIASA RR74-19, International Institute for Applied Systems Analysis, Laxenburg, Austria (1974). (2) R. Schlaifer, Probability and Statistics for Business Decisions, McGraw-Hill, New York (1959).