Rough Sets, Pareto Optimality, and Swarm Algorithms Explained

Posted on Jan 2, 2026 in Civil Engineering Technology

24(a) Meaning of Indiscernibility

• Two objects are indiscernible if they have same values for a chosen set of attributes.

• Forms equivalence classes (granules).
  

• Basis for lower/upper approximations in Rough Sets.
  

24(b) Reduct & Core

Reduct

• Minimum subset of attributes giving same classification power as full set.

• Removing any attribute → information loss.
  

Core

• Intersection of all reducts

  
  

• Contains attributes must for classification.

Example (Medical dataset)

• Attributes: Headache, Muscle Pain, Temperature → Decision: Flu

• If removing Muscle Pain still classifies correctly → {Headache, Temperature} is a reduct.
  

• If Headache appears in all reducts → Headache is in core.
  

24(c) Pareto Optimal Solution

• A solution is Pareto optimal if no other solution improves one objective without worsening another.
  

• Output:
  Set of best trade-offs, not one best point.

25(a) Dominated Set

• Solution A dominates B if:
  ✔ A is no worse in all objectives
  ✔ A is better in at least one

• Dominated set = inferior solutions → usually discarded.

25(b) ACO (Ant Colony Optimization)

• Inspired by ants leaving pheromone trails.
  

• Steps:

  1. Construct solutions using pheromone + heuristic

  2. Update pheromone (good paths reinforced)

  3. Evaporation (avoid local optimum)

• Used for TSP, routing, path optimization.
  

25(c) MOOP Example

• Multiple conflicting objectives.

• Example (Car Design):

  • Max speed

  • Min fuel

  • Min cost

• Trade-offs → Pareto front.
  

26(a) Rough Set vs Classical Set

Classical

• Crisp boundary: x ∈ A or x ∉ A.

Rough Set

• Lower approx = definitely in

• Upper approx = possibly in

• Boundary region = uncertainty.

26(b) MOOP Mathematical Form

Min/Max: F(x) = [f₁(x), f₂(x), …, fₖ(x)]
Subject to:

• gⱼ(x) ≤ 0

• hₗ(x) = 0

• Lᵢ ≤ xᵢ ≤ Uᵢ

• k ≥ 2 objectives.

26(c) Indiscernibility Relation

• IND(B): objects identical on attributes B.
  

• Creates equivalence classes.
  

• Basis for approximations and roughness.

27(a) Decision Table

Contains:

• U:

  
  Objects

• C:

  
  Condition attributes

• D:

  
  Decision attributes
  Used for classification & reduct computation.

27(b) MOOP vs Single-Objective

Single objective:

• One optimum

• Total ordering

MOOP:

• Many conflicting objectives

• Pareto set

• Partial ordering

27(c) Advantages of MOGA

• Finds entire Pareto set in one run

• Handles non-convex fronts

• No need for weights

• Maintains diversity

(a) Basic Working Steps of MOGA

1. Initialize population

2. Evaluate objectives

3. Non-dominated sorting

4. Assign fitness (crowding distance)

5. Selection

6. Crossover + mutation

7. Elitism

8. Repeat till convergence

(b) Non-Dominated Set vs Pareto Front

• Non-Dominated Set:

  
  solutions in decision space not dominated

• Pareto Front:

  
  their objective values in objective space

(c) Lower vs Upper Approximation

• Lower:

  
  granules fully inside the set (definitely in)

• Upper:

  
  granules intersect the set (possibly in)

• Boundary = Upper – Lower

29(a) Decision Table → Decision Matrix

• Matrix M(i, j) = set of attributes where objects i and j differ

• Used to find discernibility and reducts

29(b) PSO (Particle Swarm Optimization)

Particle updates velocity using:

• Inertia (current motion)

• Personal best (pbest)

• Global best (gbest)

Velocity update:
v = wv + c₁rand(pbest − x) + c₂rand(gbest − x)

30(a) Crisp vs Fuzzy vs Rough

30(b) Reduct & Core

• Reduct = minimal attribute subset for same classification

• Core = attributes in all reducts

30(c) Applications of MOOP

• Engineering design

• Portfolio optimization

• Supply chain optimization

✅ QUESTION 31(c)

Rough Sets in Feature Selection

• Compute reducts → smallest attribute sets maintaining classification

• Core + reducts → remove irrelevant features

• Helps in dimensionality reduction