You are tasked with creating a fuzzy rule-based system for Environment New Zealand, who are interested in being able to predict the current fire danger given a set of conditions. The model operates in two stages: the first stage predicts the amount of available biomass for burning (i.e. the amount of dry vegetation on the ground), and then uses this in combination with temperature to predict fire danger. The details are:

1. Available biomass (low, moderate, high) is inferred from vegetation density (which is measured from aerial photographs as vegetation mass in kg per km2) and days since last rain, where the available biomass increases as the days since last rain increases.

2. Fire danger (low, moderate, high) is based on available biomass and temperature (higher temperature increases the fire danger). Examples of the relationships between these variables are shown in the table below.

(a) Define the fuzzy membership functions (MF) and fuzzy rules for the system. You do not need to include all of the rules for the system, however include at least 2 rules for available biomass and two rules for fire danger. Include with your answer a comment regarding the form of the scale (x-axis) used with the available biomass membership function.

(b) Based on your example MFs and rules, show the steps involved in producing a valuation for the first entry in the table above. Include a description of fuzzification, rule application and defuzzification for your example.

(c) Briefly describe how fuzzy membership functions and fuzzy rules could be automatically constructed from a large set of data such as that given in the table above.