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

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/mocks/paper-final-2025/q3b

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/mocks/paper-final-2025--q3b/

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

Swipe across the diagram to read its labels ↔

![The shaded upper envelope counts any overlap only once.](https://deep-learning-study-room.vercel.app/is2/fig/v2_loan_aggregate.2c5ffa32a7.svg)

Diagram: The shaded upper envelope counts any overlap only once.. Labels: Chosen triangular output sets and their max-aggregate | 0 | 0.5 | 1 | 1 | 25 | 50 | 75 | 100 | μ | Output score | Reject | Consider | Approve | centroid ≈ 42.78

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.