N-Monitoring

Description

The $N$-monitoring problem is described in [1], sections 4.1 and 6.1.

Influence Diagram

The incluence diagram of generalized $N$-monitoring problem where $N≥1$ and indices $k=1,...,N.$ The nodes are associated with states as follows. Load state $L=\{high, low\}$ denotes the load, report states $R_k=\{high, low\}$ report the load state to the action states $A_k=\{yes, no\}$ which decide whether to fortificate failure state $F=\{failure, success\}.$ Finally, the utility at target $T$ depends on the whether $F$ fails and the fortification costs.

We draw the magnitude and cost of fortification $c_k∼U(0,1)$ from a uniform distribution. Fortification is defined

\[f(A_k=yes) = c_k\]
\[f(A_k=no) = 0\]

The probability that the load is high. We draw $x∼U(0,1)$ from uniform distribution.

\[ℙ(L=high)=x\]

The probabilities of the report states correspond to the load state. We draw the values $x∼U(0,1)$ and $y∼U(0,1)$ from uniform distribution.

\[ℙ(R_k=high∣L=high)=\max\{x,x-1\}\]
\[ℙ(R_k=low∣L=low)=\max\{y,y-1\}\]

The probabilities of failure which are decresead by fortifications. We draw the values $z∼U(0,1)$ and $w∼U(0,1)$ from uniform distribution.

\[ℙ(F=failure∣A_N,...,A_1,L=high)=\frac{\max{\{z, 1-z\}}}{\exp(∑_{k=1,...,N} f(A_k))}\]
\[ℙ(F=failure∣A_N,...,A_1,L=low)=\frac{\min{\{w, 1-w\}}}{\exp(∑_{k=1,...,N} f(A_k))}\]

Utility from consequences at target $T$ from failure state $F$

\[Y(F=failure) = 0\]
\[Y(F=success) = 100\]

Utility from consequences at target $T$ from action states $A_k$

\[Y(A_k)=-f(A_k)\]

Total utility at target $T$

\[Y(F,A_N,...,A_1)=Y(F)+∑_{k=1,...,N} Y(A_k).\]

References

  • 1Salo, A., Andelmin, J., & Oliveira, F. (2019). Decision Programming for Multi-Stage Optimization under Uncertainty, 1–35. Retrieved from http://arxiv.org/abs/1910.09196