# Q4(a) Fuzzy quality-control design [10]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-2024-final/q4a

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-2024-final--q4a/

Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.

**Question. Q4(a)** [10 marks · 2024-04-17]

Design a fuzzy logic-based quality-control system for manufacturing that detects defects and decides whether to reject or accept a product. Inputs: Pressure [0,100] and Quality of material [0,100]. Output: Defect detection [0,100]. Use three descriptors per input and five per output; draw membership functions and derive rules. If pressure and material quality are high, accept; vice versa reject. Show defuzzification for Pressure 40 and Quality 80.

### Answer

#### Given

Pressure=40, Material quality=80; ranges 0–100; three input terms and five output terms.

#### To find

A complete controller: membership graphs, rules, inference and crisp output.

#### Formula and why it applies

Fuzzification translates measured values into degrees. Min implements AND, max combines rules, and centroid converts the fuzzy output to one numerical command.



$$
\alpha_r=\min(\mu_{A_r}(x),\mu_{B_r}(y)),\quad z^*=\frac{\int z\mu_{out}(z)\,dz}{\int\mu_{out}(z)\,dz}
$$



#### Solution

**Source / assumption note.** This is a design question: membership coordinates, full rule table and decision threshold are not prescribed. The following is one explicitly specified valid Mamdani design, not the only possible numerical output. Never copy its result if you choose different triangles or rules.

##### Step 1. Specify the inputs and output

Pressure and Material quality each span 0–100. Use low=left shoulder (0,0,50), medium=triangle (0,50,100), high=right shoulder (50,100,100). Output is defect severity on 0–100; larger means worse quality.

Swipe across the diagram to read its labels ↔

![Explicit design assumption: overlapping low, medium and high input memberships, covering the full 0–100 range.](https://deep-learning-study-room.vercel.app/is2/fig/legacy24-quality-inputs.c512a5765f.svg)

Diagram: Explicit design assumption: overlapping low, medium and high input memberships, covering the full 0–100 range.. Labels: Explicit design assumption: overlapping low, medium and high input memberships, covering the full 0–100 range. | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 0 | 0.25 | 0.5 | 0.75 | 1 | Low | Medium | High | μ(x)

Explicit design assumption: overlapping low, medium and high input memberships, covering the full 0–100 range.

Swipe across the diagram to read its labels ↔

![Five assumed output descriptors. Endpoint descriptors are shoulders, so 0 and 100 remain covered.](https://deep-learning-study-room.vercel.app/is2/fig/legacy24-quality-outputs.e8de6dab72.svg)

Diagram: Five assumed output descriptors. Endpoint descriptors are shoulders, so 0 and 100 remain covered.. Labels: Five assumed output descriptors. Endpoint descriptors are shoulders, so 0 and 100 remain covered. | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 0 | 0.25 | 0.5 | 0.75 | 1 | Very low defect | Low defect | Medium defect | High defect | Very high defect | μ(x)

Five assumed output descriptors. Endpoint descriptors are shoulders, so 0 and 100 remain covered.

##### Step 2. Define all nine IF–THEN rules

| Pressure / Material quality | Low | Medium | High |
| --- | --- | --- | --- |
| Low | Very high defect | High defect | Medium defect |
| Medium | High defect | Medium defect | Low defect |
| High | Medium defect | Low defect | Very low defect |

Higher pressure and better material reduce defect severity, as stipulated in this question. This is an exam design assumption, not a universal manufacturing rule.

##### Step 3. Fuzzify the two crisp readings

| Input | Low membership | Medium membership | High membership |
| --- | --- | --- | --- |
| Pressure=40 | 1/5 | 4/5 | 0 |
| Material quality=80 | 0 | 2/5 | 3/5 |

For Pressure=40: low=max(0,(50−40)/50)=0.2; medium=40/50=0.8; high=max(0,(40−50)/50)=0. For Material quality=80: low=0, medium=(100−80)/50=0.4, high=(80−50)/50=0.6.

##### Step 4. Fire every active rule with minimum

| Pressure | Material quality | AND strength | Output descriptor |
| --- | --- | --- | --- |
| Low | Medium | min(0.2, 0.4) = 0.2 | High defect |
| Low | High | min(0.2, 0.6) = 0.2 | Medium defect |
| Medium | Medium | min(0.8, 0.4) = 0.4 | Medium defect |
| Medium | High | min(0.8, 0.6) = 0.6 | Low defect |

The other rules have a zero antecedent membership and therefore fire at zero.

##### Step 5. Aggregate rules and clip each output membership

| Descriptor | Maximum firing strength |
| --- | --- |
| Very low defect | 0.0 |
| Low defect | 0.6 |
| Medium defect | 0.4 |
| High defect | 0.2 |
| Very high defect | 0.0 |



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



Swipe across the diagram to read its labels ↔

![Clipped consequents are combined using maximum; overlapping areas are counted once for the centroid.](https://deep-learning-study-room.vercel.app/is2/fig/legacy24-quality-aggregate.dae0d30118.svg)

Diagram: Clipped consequents are combined using maximum; overlapping areas are counted once for the centroid.. Labels: Clipped consequents are combined using maximum; overlapping areas are counted once for the centroid. | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 0 | 0.25 | 0.5 | 0.75 | 1 | Aggregated output | μ(x)

Clipped consequents are combined using maximum; overlapping areas are counted once for the centroid.

##### Step 6. Calculate centroid area and first moment

Split the upper outline at every corner and intersection. On a segment [a, b] with endpoint heights u and v, integrate the straight line. This counts overlap once.



$$
A=(b-a)\frac{u+v}{2},\qquad M=aA+\frac{(b-a)^2(u+2v)}6,\qquad z^*=\frac{\sum M}{\sum A}
$$



| Segment [a, b] | Heights [u, v] | Area A | Moment M |
| --- | --- | --- | --- |
| [0, 15] | [0, 0.6] | 4.500000 | 45.000000 |
| [15, 25] | [0.6, 0.6] | 6.000000 | 120.000000 |
| [25, 35] | [0.6, 0.6] | 6.000000 | 180.000000 |
| [35, 40] | [0.6, 0.4] | 2.500000 | 93.333333 |
| [40, 50] | [0.4, 0.4] | 4.000000 | 180.000000 |
| [50, 55] | [0.4, 0.4] | 2.000000 | 105.000000 |
| [55, 65] | [0.4, 0.4] | 4.000000 | 240.000000 |
| [65, 70] | [0.4, 0.2] | 1.500000 | 100.833333 |
| [70, 75] | [0.2, 0.2] | 1.000000 | 72.500000 |
| [75, 95] | [0.2, 0.2] | 4.000000 | 340.000000 |
| [95, 100] | [0.2, 0] | 0.500000 | 48.333333 |



$$
z^*=\frac{1525.000000}{36.000000}\approx 42.361111
$$



##### Step 7. Make the accept/reject decision

Choose and state a threshold: accept when defect severity is below 50, reject at or above 50. The calculated output is below 50, so accept. High/high gives very low defects and low/low gives very high defects, satisfying the question.

**Final answer.** With the stated membership functions and rules, the output is 42.361111 on the given 0–100 scale. Under the stated threshold of 50: accept.