Intelligent Systems II · M2 · papers

Q2(a) Fuzzy fire-hazard system [10]

Re-examination · 31 Jan 2026

Longer past-paper practice restricted to Fuzzy, GA and PSO.

A building safety department wants a fuzzy logic system to predict fire hazards. Inputs are Temperature (1–100°C) and Smoke Level (1–100). The output Fire Risk (1–100) has labels Low, Moderate and High. Define triangular membership functions, construct at least nine rules, explain Mamdani inference and defuzzify for Temperature=65°C and Smoke=80. [10 marks]

Answer

Given

Variable	Crisp value
Temperature	65
Smoke Level	80

The question asks us to design the membership functions. This solution chooses Low=Tri(1,25,50), Medium=Tri(25,50,75), High=Tri(50,75,100) for both inputs. Output labels use the same triangles in their respective order. These are explicit design choices, not additional printed givens.

To find

A complete stated design and its continuous Mamdani centroid.

Formula and why it applies

Use min for AND and implication, max for aggregation and the true area centroid of the resulting shape. A peak-only weighted average is a different defuzzifier.

z^*=\frac{\int z\mu_{out}(z)\,dz}{\int\mu_{out}(z)\,dz}

Solution

Step 1. Specify the membership functions

Input / output position	Triangle	Peak
Low / Low	(1,25,50)	25
Medium / Moderate	(25,50,75)	50
High / High	(50,75,100)	75

Use 0 outside each triangle, the rising fraction before the peak and the falling fraction after it. This design is evaluated at the requested input; a deployed controller would also need coverage/fallback at all boundary values.

Step 2. Write all nine rules

Temperature / Smoke Level	Low	Medium	High
Low	Low	Low	Moderate
Medium	Low	Moderate	High
High	Moderate	High	High

Read each cell as an IF–AND–THEN rule. For example, the centre cell combines Medium with Medium.

Step 3. Fuzzify Temperature = 65

\mu_{Low}(65)=0

\mu_{Medium}(65)=\frac{75-65}{75-50}=0.4

\mu_{High}(65)=\frac{65-50}{75-50}=0.6

Step 4. Fuzzify Smoke Level = 80

\mu_{Low}(80)=0

\mu_{Medium}(80)=0

\mu_{High}(80)=\frac{100-80}{100-75}=0.8

Step 5. Evaluate all rules and aggregate matching consequents

Temperature	Smoke Level	Firing strength	Consequent
Low	Low	min(0,0) = 0	Low
Low	Medium	min(0,0) = 0	Low
Low	High	min(0,0.8) = 0	Moderate
Medium	Low	min(0.4,0) = 0	Low
Medium	Medium	min(0.4,0) = 0	Moderate
Medium	High	min(0.4,0.8) = 0.4	High
High	Low	min(0.6,0) = 0	Moderate
High	Medium	min(0.6,0) = 0	High
High	High	min(0.6,0.8) = 0.6	High

Aggregated label strengths: Low = 0, Moderate = 0, High = 0.6.

\mu_{out}(z)=\max_r\min(\alpha_r,\mu_{C_r}(z))

Step 6. Draw the aggregated output

The shaded upper envelope counts any overlap only once.

Step 7. Calculate area and moment of each straight-line strip

For a strip from a to b with endpoint heights u and v, its trapezoid area is A. Its first moment M is the integral of z times the line height.

A=(b-a)(u+v)/2

M=\frac{b-a}{6}\big[(2a+b)u+(a+2b)v\big]

a	b	u	v	Area A	Moment M
50	65	0	0.6	4.5	270
65	85	0.6	0.6	12	900
85	100	0.6	0	4.5	405

Step 8. Divide total moment by total area

\sum A=21

\sum M=1575

z^*=\frac{1575}{21}\approx75

Final answer. For these stated triangles and rules, the centroid is 75. This is in the high-risk region of this illustrative fuzzy design.

Because the paper asks for a design, a different explicitly specified membership/rule system can yield a different numerical output. Explain the choices and use the chosen defuzzification method consistently; no particular examiner mark outcome is promised.

Why the clipped High shape is symmetric about 75

Only the High output survives, at height .6. Its triangle is (50,75,100). Solve its rising edge: (z−50)/25=.6, so z=50+15=65. Solve its falling edge: (100−z)/25=.6, so z=100−15=85.

Piece	Area calculation	Centre	Moment
Left triangle [50,65]	½×15×.6=4.5	(50+65+65)/3=60	4.5×60=270
Rectangle [65,85]	20×.6=12	(65+85)/2=75	12×75=900
Right triangle [85,100]	½×15×.6=4.5	(85+85+100)/3=90	4.5×90=405

The horizontal centroid of a triangle is the average of the three vertex x-coordinates. Add areas: 4.5+12+4.5=21. Add moments:270+900+405=1575. Divide 1575/21=75, agreeing with symmetry.