🌿📉 Mathematical Modeling of UFLP Using Binary Grasshopper Swarms | #Sciencefather #researchers #algorithm

🧮🦗 A Binary Grasshopper Optimization Algorithm for Discrete Mathematical Modeling in UFLP

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🔷 1. Introduction: The Mathematical Challenge

The Uncapacitated Facility Location Problem (UFLP) is a classic in operations research, where the objective is to determine:

• Which facilities to open, and

• How to assign each customer to exactly one facility,
  such that the total cost (facility opening + service costs) is minimized.

Let:

• $F = \{1,2,\dots,m\}$: potential facility sites

• $C = \{1,2,\dots,n\}$: customer locations

• $f_i$: cost to open facility $i$

• $c_{ij}$: cost to serve customer $j$ from facility $i$

Objective function:

$$\textbf{Minimize } Z = \sum_{i=1}^{m} f_i x_i + \sum_{i=1}^{m} \sum_{j=1}^{n} c_{ij} y_{ij}$$

Subject to constraints:

• Each customer assigned once:

  $$\sum_{i=1}^{m} y_{ij} = 1 \quad \forall j$$

• Assignments only from open facilities:

  $$y_{ij} \leq x_i \quad \forall i, j$$

• Binary variables:

  $$x_i, y_{ij} \in \{0, 1\}$$

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🧠 2. Nature-Inspired Intelligence: The Grasshopper's Strategy

The Grasshopper Optimization Algorithm (GOA) simulates the swarming behavior of grasshoppers. The mathematical model includes:

$$X_i = \sum_{j=1}^{N} s(d_{ij}) \cdot \hat{d}_{ij} + G + W$$

Where:

• $s(d)$ = social interaction force

• $G$ = gravity force

• $W$ = wind influence

This model is continuous, but we apply a binary transformation to adapt it to combinatorial optimization problems like UFLP.

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🔁 3. Binary Transformation: Sigmoid Discretization

To convert real-valued solutions into binary form, we apply a sigmoid transfer function:

$$T(X) = \frac{1}{1 + e^{-X}}$$

Then use a threshold:

$$x_i^{t+1} = \begin{cases} 1 & \text{if } rand() < T(X_i^t) \\ 0 & \text{otherwise} \end{cases}$$

This maps the grasshopper's real-valued position into binary — indicating whether a facility is open (1) or closed (0).

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🎯 4. Fitness Function: Evaluating the Binary Solution

Each binary vector $\mathbf{x}$ encodes facility decisions. The fitness is:

$$f(\mathbf{x}) = \sum_{i=1}^{m} f_i x_i + \sum_{j=1}^{n} \min_{\substack{i=1\\x_i=1}}^m c_{ij}$$

To handle constraint violations, a penalty function is added:

$$f_{\text{penalized}}(\mathbf{x}) = f(\mathbf{x}) + \lambda \cdot V(\mathbf{x})$$

Where $V(\mathbf{x})$ counts constraint violations and $\lambda$ is the penalty weight.

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⚙️ 5. Algorithmic Workflow: Step-by-Step

Step	Description
1. Initialization	Generate random binary solutions (grasshopper swarm)
2. Evaluation	Compute fitness using cost function
3. Position Update	Apply GOA movement equations and binary mapping
4. Repair	Fix constraint violations or penalize them
5. Iteration	Repeat until stopping criteria are met
6. Output	Return the best-found binary solution

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✨ 6. Why It Works: Mathematical Elegance Meets Natural Design

• ✅ Binary Logic meets Biological Motion

• ✅ Efficient in large, high-dimensional spaces

• ✅ Flexible and robust in real-world logistics

• ✅ Aesthetic fusion of swarm intelligence and combinatorial math

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💭 Final Remark

The Binary Grasshopper Optimization Algorithm offers a visually intuitive, mathematically sound, and biologically inspired approach to solving the UFLP. It bridges nature and mathematics in a dynamic optimization framework — where every grasshopper's leap echoes a decision in facility planning.

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