Optimally configuring a measurement system to detect diversions from a nuclear fuel cycle

Abstract

The civilian nuclear fuel cycle is an industrial process that produces electrical power from the nuclear fission of uranium. Using a measurement system to accurately account for possibly dangerous nuclear material, such as uranium, in a fuel cycle is important because of its possible loss or diversion. A measurement system is defined by a set of measurement methods, or “devices,” used to account for material flows and inventory values at specific locations at facilities in the fuel cycle. We develop a simulation-optimization algorithm and an integer-programming model to find the best, or near-best, resource-limited measurement system with a high degree of confidence. The simulation-optimization algorithm minimizes a weighted sum of false positive and false negative diversion-detection probabilities while accounting for material quantities and measurement errors across a finite, discrete time horizon in hypothetical non-diversion and diversion contexts. In each time period, the estimated cumulative material unaccounted for is compared to a fixed or an optimized threshold value to assess if a “significant amount of material” is lost from a measurement system. The integer-programming model minimizes the population variance of the estimated material loss, i.e., material unaccounted for, in a measurement system. We analyze three potential problems in nuclear fuel cycle measurement systems: (i) given location-dependent device precisions, find the configuration of n devices at n locations ($n=3$) that provides the lowest corresponding objective values using the simulation-optimization algorithm and integer-programming model, (ii) find the location at which improving device precision reduces objective values the most using the simulation-optimization algorithm (given the device accuracy is 100%), and (iii) determine the effect of measurement frequency on measurement system configurations and objective values using the simulation-optimization algorithm. We obtain comparable results for each problem at least an order of magnitude faster than existing methods do. Using an optimized, rather than a fixed, detection threshold in the simulation-optimization algorithm reduces the weighted sum of false positive and false negative probabilities.

Appendix: MVCM properties

In this appendix, we derive the properties of the MVCM formulation for a single key measurement point. We begin by considering a continuous time ${\tilde{C}}_t$ modeled by a Gaussian Process.

Type I error case:

Let ${\tilde{C}}_{1t}$ be a Gaussian Process without drift, and $\hbox {Var}({\tilde{C}}_{1,t+\tau }-{\tilde{C}}_{1t})=\sigma _1^2 \tau$ with independent increments. Let $T_{\underline{{\lambda }}}$ be the first time ${\tilde{C}}_{1t}={\underline{{\lambda }}}$.

Let ${\tilde{C}}_{2t}$ be a Gaussian Process without drift, and $\hbox {Var}({\tilde{C}}_{2,t+\tau }-{\tilde{C}}_{2t})=\sigma _2^2 \tau$ with independent increments where $\sigma _2^2<\sigma _1^2$. Let $S_{\underline{{\lambda }}}$ be the first time ${\tilde{C}}_{2t}={\underline{{\lambda }}}$.

$$\begin{aligned} P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}})&= P({\tilde{C}}_{1t} \ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}\le t)P(T_{\underline{{\lambda }}}\le t)+P({\tilde{C}}_{1t} \ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}>t)P(T_{\underline{{\lambda }}}>t) \end{aligned}$$

(40)

$$\begin{aligned}&\text {note } P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}\le t) =\frac{1}{2} \text { and } P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}>t) = 0 \end{aligned}$$

(41)

$$\begin{aligned}&\Rightarrow P(T_{\underline{{\lambda }}}\le t)=2 P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}) \end{aligned}$$

(42)

Similarly,

$$\begin{aligned} P(S_{\underline{{\lambda }}}\le t)= 2 P({\tilde{C}}_{2t} \ge {\underline{{\lambda }}}) \end{aligned}$$

(43)

Now, we have:

$$\begin{aligned} P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}})&= \int _{{\underline{{\lambda }}}/(\sqrt{t} \sigma _1)}^\infty \frac{1}{\sqrt{2 \pi \sigma _1^2t}} \exp (-\frac{x^2}{2\sigma _1^2t})dx \nonumber \\&> \int _{{\underline{{\lambda }}}/(\sqrt{t} \sigma _2)}^\infty \frac{1}{\sqrt{2 \pi \sigma _2^2t}} \exp (-\frac{y^2}{2\sigma _2^2t})dy=P({\tilde{C}}_{2t}\ge {\underline{{\lambda }}}) \end{aligned}$$

(44)

Therefore, reducing the variance at a single key measurement point reduces the Type I error probability for a device at that key measurement point before any time t.

Type II error case:

Let ${\tilde{C}}_{1t}$ be a Gaussian Process with drift ${ C}_{t}$, and $\hbox {Var}({\tilde{C}}_{1,t+\tau }-{\tilde{C}}_{1t})=\sigma _1^2 \tau$ with independent increments. Let $T_{\underline{{\lambda }}}$ be the first time ${\tilde{C}}_{1t}={\underline{{\lambda }}}$.

Let ${\tilde{C}}_{2t}$ be a Gaussian Process with drift ${ C}_{t}$, and $\hbox {Var}({\tilde{C}}_{2,t+\tau }-{\tilde{C}}_{2t})=\sigma _2^2 \tau$ with independent increments where $\sigma _2^2<\sigma _1^2$. Let $S_{\underline{{\lambda }}}$ be the first time ${\tilde{C}}_{2t}={\underline{{\lambda }}}$.

$$\begin{aligned} P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}})&= P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}\le t)P(T_{\underline{{\lambda }}}\le t)+P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}|T_{\underline{{\lambda }}}>t)P(T_{\underline{{\lambda }}}>t) \end{aligned}$$

(45)

$$\begin{aligned}&\Rightarrow P({\tilde{C}}_{1t} - { C}_{t}\ge {\underline{{\lambda }}}-{ C}_{t} |T_{\underline{{\lambda }}}\le t)=\int _{\underline{{\lambda }}}^\infty \frac{1}{\sqrt{2\pi \sigma _1^2t}\exp (-\frac{(x-C_{t})^2}{2\sigma _1^2t})} \end{aligned}$$

(46)

$$\begin{aligned}&\Rightarrow P(T_{\underline{{\lambda }}}\ge t)=\frac{1}{\int _{\underline{{\lambda }}}^\infty \frac{1}{\sqrt{2\pi \sigma _1^2t}}\exp (-\frac{(x-{ C}_{t})^2}{2\sigma _1^2t})} P({\tilde{C}}_{1t}\ge {\underline{{\lambda }}}) \end{aligned}$$

(47)

Similarly,

$$\begin{aligned} P(S_{\underline{{\lambda }}}\ge t)= \frac{1}{\int _{\underline{{\lambda }}}^\infty \frac{1}{\sqrt{2\pi \sigma _2^2t}}\exp (-\frac{(y-{ C}_{t})^2}{2\sigma _2^2t})dx} P({\tilde{C}}_{2t} \ge {\underline{{\lambda }}}) \end{aligned}$$

(48)

Now, we have:

$$\begin{aligned} P(T_{\underline{{\lambda }}}\ge t)=\frac{\int _{{\underline{{\lambda }}}/(\sqrt{t}\sigma _1)}^\infty \frac{1}{\sqrt{2 \pi \sigma _1^2t}} \exp (-\frac{(x-{ C}_{t})^2}{2\sigma _1^2t})dx}{\int _{\underline{{\lambda }}}^\infty \frac{1}{\sqrt{2\pi \sigma _1^2t}} \exp (-\frac{(x-{ C}_{t})^2}{2\sigma _1^2t})dx} \end{aligned}$$

(49)

which is not uniformly related to

$$\begin{aligned} \frac{\int _{{\underline{{\lambda }}}/(\sqrt{t}\sigma _2)}^\infty \frac{1}{\sqrt{2 \pi \sigma _2^2t}} \exp (-\frac{(y-{ C}_{t})^2}{2\sigma _2^2t})dy}{\int _{\underline{{\lambda }}}^\infty \frac{1}{\sqrt{2\pi \sigma _2^2t}}\exp ( -\frac{(y-C_{t})^2}{2\sigma _2^2t})dy} =P(S_{\underline{{\lambda }}}\ge t) \end{aligned}$$

(50)

This result implies no guarantee that the Type II error probability at a given key measurement point before any time t is lowered by reducing the measurement error at that key measurement point. Therefore, we can conclude that lowering the measurement error generating ${\tilde{C}}_t$ provides a lower Type I error probability at a given key measurement point before any time t, but may not lower the Type II error probability. However, we note that the Type II error probability is often lowered in SOCA simulations. Because these results hold for every t denoting continuous time, they also hold for ${\tilde{C}}_t$ where $t=1,\ldots ,T$, indexing discrete time by means of integration with respect to t. Finally, these results are suggestive of a general pattern for T large enough, as ${\tilde{C}}_t$ will converge to a Gaussian random variable as $t \rightarrow \infty$ by the Central Limit Theorem.