# Q2(a) Fuzzy investment-controller design and centroid [10]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-2025-april/q2a

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-2025-april--q2a/

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

**Question. Q2(a)** [10 marks · 2025-04-28]

Create a fuzzy logic system to assist in investment decision-making. The system should consider variables such as risk tolerance, and historical performance to provide recommendations on asset allocation and investment strategies.

Input variables: Risk Tolerance (range 1-100) and historical performance (range 1-100). Output variable is Asset allocation (range 1-100) (Conservative, Balanced and Aggressive). Use 3 descriptors for input as well as output. Set rules to accurately predict the output. Show defuzzification for risk 40 and historical performance 8.

### Answer

### Given

**Printed inputs:** risk=40, historical performance=8, both on 1–100; output allocation score on 1–100. Membership shapes, breakpoints, rules and inference method are design choices, not printed data.

### To find

One fully specified fuzzy system and the crisp output for these inputs.

### Formula and why

Use a Mamdani system: fuzzify inputs → apply min for AND → clip each output membership at its rule strength → aggregate by max → centroid. The result is a defined model output, not a guaranteed real investment return.



$$
z^*=\frac{\int_1^{100}z\mu_{agg}(z)\,dz}{\int_1^{100}\mu_{agg}(z)\,dz}
$$



### Step 1. Specify all membership functions

For either input t∈[1,100], choose Low as a left shoulder, Medium as a triangle and High as a right shoulder:



$$
\mu_L(t)=\begin{cases}(50-t)/49&1\le t\le50\\0&50<t\le100\end{cases}
$$





$$
\mu_M(t)=\begin{cases}(t-1)/49&1\le t\le50\\(100-t)/50&50<t\le100\end{cases}
$$





$$
\mu_H(t)=\begin{cases}0&1\le t\le50\\(t-50)/50&50<t\le100\end{cases}
$$



Use the same three shapes on output z, named Conservative, Balanced and Aggressive. The score is an increasing aggressiveness index; it is not automatically a percentage in one specified asset. Higher scores recommend a more aggressive strategy within this illustrative model.

Swipe across the diagram to read its labels ↔

![Declared membership design. Both inputs and the output use the same breakpoints 1,50,100, with different linguistic labels.](https://deep-learning-study-room.vercel.app/is2/fig/legacy25-memberships.904ed2cacf.svg)

Diagram: Declared membership design. Both inputs and the output use the same breakpoints 1,50,100, with different linguistic labels.. Labels: Declared shoulder triangular and shoulder membership functions | Input value / allocation score | μ | 0 | 1 | 1 | 50 | 100 | Low / Conservative | Medium / Balanced | High / Aggressive

Declared membership design. Both inputs and the output use the same breakpoints 1,50,100, with different linguistic labels.

### Step 2. State the nine rules

| Risk tolerance | Historical performance | Output |
| --- | --- | --- |
| Low | Low | Conservative |
| Low | Medium | Conservative |
| Low | High | Conservative |
| Medium | Low | Conservative |
| Medium | Medium | Balanced |
| Medium | High | Balanced |
| High | Low | Balanced |
| High | Medium | Aggressive |
| High | High | Aggressive |

The rule base respects low risk tolerance even if past performance is high. Medium risk plus poor past performance moves towards Conservative. High risk with medium/high performance allows Aggressive. These are declared illustrative design judgments; other valid designs can yield other outputs.

### Step 3. Fuzzify the two printed numbers

| Input | Low | Medium | High |
| --- | --- | --- | --- |
| Risk 40 | (50−40)/49=10/49=0.204082 | (40−1)/49=39/49=0.795918 | 0 |
| Performance 8 | (50−8)/49=42/49=0.857143 | (8−1)/49=7/49=1/7=0.142857 | 0 |

**8 is retained exactly; it is not replaced with 80.**

### Step 4. Evaluate all nonzero rules

| Rule | AND calculation | Firing strength | Consequent |
| --- | --- | --- | --- |
| Low & Low | min(10/49,42/49) | 10/49 | Conservative |
| Low & Medium | min(10/49,7/49) | 7/49 | Conservative |
| Medium & Low | min(39/49,42/49) | 39/49 | Conservative |
| Medium & Medium | min(39/49,7/49) | 7/49 | Balanced |

The other five rules have a zero antecedent and do not fire. Maximum aggregation gives Conservative height 39/49, Balanced height 1/7 and Aggressive height 0.

### Step 5. Construct the aggregated output



$$
\mu_{agg}(z)=\max\{\min(39/49,\mu_L(z)),\min(1/7,\mu_M(z))\}
$$



The Conservative clip meets its descending slope at z=11. It falls to 1/7 at z=43. The Balanced descending side reaches 1/7 at z=650/7≈92.857143.



$$
\mu_{agg}(z)=\begin{cases}39/49&1\le z\le11\\(50-z)/49&11<z\le43\\1/7&43<z\le650/7\\(100-z)/50&650/7<z\le100\end{cases}
$$



Swipe across the diagram to read its labels ↔

![Actual max-aggregated output for risk 40 and performance 8. The shaded area has centroid 30.379571.](https://deep-learning-study-room.vercel.app/is2/fig/legacy25-centroid.6d2be43cdb.svg)

Diagram: Actual max-aggregated output for risk 40 and performance 8. The shaded area has centroid 30.379571.. Labels: Mamdani aggregated output and centroid for risk40 history8 | Input value / allocation score | μ | 0 | 1 | 1 | 50 | 100 | Centroid z\* =30.379571

Actual max-aggregated output for risk 40 and performance 8. The shaded area has centroid 30.379571.

### Step 6. Calculate area and first moment for each segment

For a segment µ(z)=mz+c between a and b, integrate its area A and first moment M as follows:



$$
A=\frac m2(b^2-a^2)+c(b-a),\qquad M=\frac m3(b^3-a^3)+\frac c2(b^2-a^2)
$$



| Interval | Membership | Area A | First moment M |
| --- | --- | --- | --- |
| 1 to 11 | 39/49 | 7.959183673 | 47.755102041 |
| 11 to 43 | (50−z)/49 | 15.020408163 | 349.823129252 |
| 43 to 650/7 | 1/7 | 7.122448980 | 483.817784257 |
| 650/7 to 100 | (100−z)/50 | 0.510204082 | 48.590864917 |

For example, the first rectangle has area (11−1)×39/49=390/49 and moment (39/49)×(11²−1²)/2=2340/49. The other areas and moments follow explicitly: interval 11–43 has A=[−z²/98+50z/49] from 11 to 43=15.020408163 and M=[−z³/147+25z²/49] from 11 to 43=349.823129252. Interval 43–650/7 has A=(650/7−43)/7=7.122448980 and M=((650/7)²−43²)/14=483.817784257. Interval 650/7–100 has A=[2z−z²/100] from 650/7 to 100=0.510204082 and M=[z²−z³/150] from 650/7 to 100=48.590864917.

### Step 7. Divide moment by area and interpret



$$
\sum A=1500/49\approx30.612244898
$$





$$
\sum M=637971/686\approx929.986880466
$$





$$
z^*=\frac{637971/686}{1500/49}=\frac{212657}{7000}\approx\boxed{30.379571}
$$



The resulting allocation score leans Conservative for this declared rule base. A centroid is obtained from the aggregated area; do not replace this Mamdani calculation with an unexplained weighted average of label names.