Risk assessment of hazardous waste situations requires the assimilation of a great deal of linguistic information. In this context, consider risk as being defined as "consequence of a hazard" multiplied by "possibility of the hazard" (instead of the conventional "probability of hazard"). Consequence is the result of an unintended action on humans, equipment, or facilities, or the environment. Possibility is the estimate of the likelihood that the unintended action will occur. Consequence and possibility are dependent on several factors and therefore cannot be determined with precision. We will use composition, then, to define risk; hence risk $=$ consequence $\circ$ possibility, or
$$
\underline{R}=\mathrm{C} \circ \mathrm{P}
$$
We will consider that the consequence is the logical intersection of the hazard mitigation and the hazard source term (the source term defines the type of initiation, such as a smokestack emitting a toxic gas, or a truck spill of a toxic organic); hence we define consequence $=$ mitigation $\cap$ source term, or
$$
\underset{\sim}{\mathrm{C}}=\underset{\sim}{\mathrm{M}} \cap \underset{\sim}{\mathrm{ST}}
$$
Since humans and their systems are ultimately responsible for preventing or causing non-natural hazards, we define the possibility of a hazard as the logical intersection of human errors and system vulnerabilities; hence possibility $=$ human factors $\cap$ system reliabilities,
$$
\underset{\sim}{\mathrm{P}}=\underset{\sim}{\mathrm{H}} \cap \underset{\sim}{\mathrm{S}}
$$
From these postulates, show that the membership form of the risk, R , is given by the expression
$$
\mu_{\mathbb{R}}(x, y)=\max \left[\min \left(\min \left[\mu_{\mathrm{M}}(x, y), \mu_{\mathrm{ST}}(x, y)\right], \min \left[\mu_{\mathrm{H}}(x, y), \mu_{\mathrm{S}}(x, y)\right]\right)\right]
$$