Intelligent Systems II · TEE · mocks

Q3(b) Fuzzy loan-approval system [10]

Final examination · 24 Nov 2025

100-mark timed practice using the actual Final 2025 paper. Q1 compulsory; any four of Q2–Q7. Worked solutions are below the questions.

A bank uses fuzzy decision support with Credit Score and Income Stability, both on 1–100. Loan Approval is a score on 1–100 with labels Reject, Consider and Approve. Define triangular membership functions, construct nine rules, explain Mamdani inference and defuzzify for Credit Score=55 and Income Stability=35. [10 marks]

Answer

Given

Variable	Crisp value
Credit Score	55
Income Stability	35

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 / Reject	(1,25,50)	25
Medium / Consider	(25,50,75)	50
High / Approve	(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

Credit Score / Income Stability	Low	Medium	High
Low	Reject	Reject	Consider
Medium	Reject	Consider	Approve
High	Consider	Approve	Approve

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

Step 3. Fuzzify Credit Score = 55

\mu_{Low}(55)=0

\mu_{Medium}(55)=\frac{75-55}{75-50}=0.8

\mu_{High}(55)=\frac{55-50}{75-50}=0.2

Step 4. Fuzzify Income Stability = 35

\mu_{Low}(35)=\frac{50-35}{50-25}=0.6

\mu_{Medium}(35)=\frac{35-25}{50-25}=0.4

\mu_{High}(35)=0

Step 5. Evaluate all rules and aggregate matching consequents

Credit Score	Income Stability	Firing strength	Consequent
Low	Low	min(0,0.6) = 0	Reject
Low	Medium	min(0,0.4) = 0	Reject
Low	High	min(0,0) = 0	Consider
Medium	Low	min(0.8,0.6) = 0.6	Reject
Medium	Medium	min(0.8,0.4) = 0.4	Consider
Medium	High	min(0.8,0) = 0	Approve
High	Low	min(0.2,0.6) = 0.2	Consider
High	Medium	min(0.2,0.4) = 0.2	Approve
High	High	min(0.2,0) = 0	Approve

Aggregated label strengths: Reject = 0.6, Consider = 0.4, Approve = 0.2.

\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
1	15.4	0	0.6	4.32	45.792
15.4	25	0.6	0.6	5.76	116.352
25	35	0.6	0.6	6	180
35	40	0.6	0.4	2.5	93.333333333
40	50	0.4	0.4	4	180
50	55	0.4	0.4	2	105
55	65	0.4	0.4	4	240
65	70	0.4	0.2	1.5	100.833333333
70	75	0.2	0.2	1	72.5
75	95	0.2	0.2	4	340
95	100	0.2	0	0.5	48.333333333

Step 8. Divide total moment by total area

\sum A=35.58

\sum M=1522.144

z^*=\frac{1522.144}{35.58}\approx42.780888139

Final answer. For these stated triangles and rules, the centroid is 42.780888139. This is a fuzzy suitability score; the category memberships can be evaluated at this score. The model alone is not a complete loan-decision policy.

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.

How the centroid table’s breakpoints are obtained

For each output triangle, solve “line height = clip height”. Reject is clipped at .6: rising endpoint 1+.6(25−1)=15.4 and falling endpoint 50−.6(50−25)=35. Consider is clipped at .4: endpoints 25+.4(50−25)=35 and 75−.4(75−50)=65. Approve is clipped at .2: endpoints 50+.2(75−50)=55 and 100−.2(100−75)=95.

Now take the maximum of the three clipped curves. The declining Reject edge reaches the Consider height .4 at (50−z)/25=.4, hence z=40. The declining Consider edge reaches the Approve height .2 at (75−z)/25=.2, hence z=70. These crossover locations must be included when splitting the aggregate. Original peaks and other clip endpoints may add extra splits without changing the total area.

For the first aggregate strip from a=1 to b=15.4, endpoint heights are u=0 and v=0.6. Find its area and first moment. [3 marks · Worked example]

Given

Strip endpoints a=1 and b=15.4; left height u=0 and right height v=0.6. This is a straight rising edge of the clipped Reject set.

To find

A and M for one table row.

Formula and why it applies

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

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

A is trapezoid area. M weights every small area by its horizontal position; this formula integrates a straight line.

Solution

Step 1. Width and height sum

b-a=15.4-1=14.4\qquad u+v=0.6

Step 2. Area

A=14.4\times0.6/2=8.64/2=4.32

Step 3. First bracket term

(2a+b)u=(2+15.4)\times0=0

Step 4. Second bracket term

(a+2b)v=(1+30.8)\times0.6=31.8\times0.6=19.08

Step 5. Moment

M=(14.4/6)(0+19.08)=2.4\times19.08=45.792

Final answer. Area=4.32; moment=45.792. Repeat the same arithmetic for each row, then sum the areas and moments separately.

Interpret the loan result using an explicit decision rule

Choose the output label having the greatest membership at the centroid; this is an additional stated interpretation rule. At z=42.780888139:

\mu_{Reject}=(50-z)/25\approx0.288764474,\quad\mu_{Consider}=(z-25)/25\approx0.711235526,\quad\mu_{Approve}=0

The resulting recommendation is Consider / manual review; this design does not recommend automatic approval. If a binary approve/reject answer is required, the question must supply a threshold or you must state the threshold you adopt. A different decision threshold is separate from defuzzification.