https://github.com/nafishr24/linear-programming

This Python project implements the isoline method to solve linear optimization problems for three constraint cases: dual ≤ constraints, dual ≥ constraints, and mixed ≤/≥ constraints, providing both computational solutions and visual representations of the optimization process.
https://github.com/nafishr24/linear-programming

linear-programming math optimization python

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This Python project implements the isoline method to solve linear optimization problems for three constraint cases: dual ≤ constraints, dual ≥ constraints, and mixed ≤/≥ constraints, providing both computational solutions and visual representations of the optimization process.

• Host: GitHub
• URL: https://github.com/nafishr24/linear-programming
• Owner: nafishr24
• License: mit
• Created: 2025-04-11T06:24:11.000Z (about 1 year ago)
• Default Branch: main
• Last Pushed: 2025-09-10T08:41:46.000Z (9 months ago)
• Last Synced: 2025-09-10T12:05:45.382Z (9 months ago)
• Topics: linear-programming, math, optimization, python
• Language: Python
• Homepage:
• Size: 624 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

Awesome Lists containing this project

README

          

# 🧮 Isoline Method in Linear Programming

## 📌 Overview

The **Isoline Method** (also known as *isoprofit* or *isocost line*) is a **graphical technique** used in **linear programming with two variables** to find the **optimal solution** of an objective function, either for **maximization** or **minimization**.

---

## 📚 1. What is the Isoline Method?

The isoline is a line representing all combinations of `x` and `y` that yield the same value of the objective function:  

```

Z = gx + hy

````

By shifting this line **parallelly**, we can determine which point in the feasible region gives the **maximum** or **minimum** value of `Z`.

---

## 🧠 2. Basic Concept

- Each isoline represents a **constant value** of the objective function.

- The lines have the **same slope** (gradient).

- The **optimal solution** is found at the point where the **isoline is tangent to the feasible region**:

  - **Maximization** → Farthest isoline from origin that still touches feasible region.

  - **Minimization** → Nearest isoline to the origin that still touches feasible region.

---

## 🪜 3. Steps to Use the Isoline Method

1. **Define the objective function and constraints**  

   Example: `Z = 5x + 3y`

2. **Plot the constraints** on the coordinate system and shade the feasible region.

3. **Draw isolines** for different values of `Z`:  

   Example: `Z = 10, 20, 30` → same slope

4. **Shift the isoline** in the optimal direction:

   - ➕ Maximization → Away from origin  

   - ➖ Minimization → Toward origin

5. **Determine the optimal point**:

   - Where the isoline last/first touches the feasible region

---

## ⚖️ 4. Advantages & Limitations

| ✅ Advantages                     | ⚠️ Limitations                             |

|----------------------------------|--------------------------------------------|

| Visual and intuitive             | Only works for 2-variable problems         |

| Easy to teach and understand     | Not scalable for large problems            |

| Ideal for educational purposes   | Needs plotting tools for precision         |

---

## 🧪 5. Analysis of Constraint Scenarios

### A. Both Constraints with ≤ Signs

```text

a₁x + b₁y ≤ c₁  

a₂x + b₂y ≤ c₂  

x, y ≥ 0

````

* Represents **limited resources**

* Often **bounded** region

* Typically used in **maximization**

---

### B. Both Constraints with ≥ Signs

```text

a₁x + b₁y ≥ c₁  

a₂x + b₂y ≥ c₂  

x, y ≥ 0

```

* Represents **minimum requirements**

* Region is usually **unbounded**

* Suitable for **minimization**

---

### C. Mixed Signs (One ≤, One ≥)

```text

a₁x + b₁y ≤ c₁  

a₂x + b₂y ≥ c₂  

x, y ≥ 0

```

* Common in **real-world problems**

* Region can be **bounded or unbounded**

* Careful analysis needed

---

## 📊 6. Summary Table

| Case               | Feasible Region   | Optimal Solution (Min) | Optimal Solution (Max)       |

| ------------------ | ----------------- | ---------------------- | ---------------------------- |

| Both ≤ Constraints | Bounded/Unbounded | Closest corner point   | Farthest corner or unbounded |

| Both ≥ Constraints | Usually Unbounded | Closest corner point   | Unbounded or none            |

| Mixed Constraints  | Bounded/Unbounded | Depends on geometry    | May be bounded/unbounded     |

---

## 💻 7. Python Implementation

This repository includes a Python implementation to **visualize the isoline method** and **solve LP problems** using matplotlib.

It supports:

* Two constraints with ≤

* Two constraints with ≥

* One constraint ≤ and one ≥

### ✅ How to Use

First, import the library:

```python

from isoline import *

```

Then, choose the function based on the constraint signs:

---

### 🔹 Case 1: Both constraints use `≥`

Use `minimize()` for **minimum requirement problems**:

```python

minimize(

    a=5, b=3, c=30,        # Constraint 1: 5x + 3y ≥ 30

    d=4, e=3, f=24,        # Constraint 2: 4x + 3y ≥ 24

    g=20000, h=16000,      # Objective Function: Z = 20000x + 16000y

    x_max=15, y_max=15     # Axis limits for graph

)

```

---

### 🔹 Case 2: Both constraints use `≤`

Use `maximize()` for **resource limitation problems**:

```python

maximize(

    a1=2, b1=1, c1=300,     # Constraint 1: 2x + y ≤ 300

    a2=1, b2=2, c2=300,     # Constraint 2: x + 2y ≤ 300

    g=150, h=100,           # Objective Function: Z = 150x + 100y

    x_max=300, y_max=300    # Axis limits for graph

)

```

---

### 🔹 Case 3: Mixed signs (`≤` and `≥`)

Use `optimize()` when constraint signs are **mixed**:

```python

optimize(

    a=4, b=3, c=24,         # Constraint 1: 4x + 3y ≤ 24

    d=2, e=5, f=20,         # Constraint 2: 2x + 5y ≥ 20

    g=7, h=5,               # Objective Function: Z = 7x + 5y

    x_max=10, y_max=8       # Axis limits for graph

)

```

Each function will:

* Display the feasible region

* Plot isolines for different Z values

* Highlight the **optimal solution**

---

## 📎 8. Installation

Make sure you have the following packages installed:

```bash

pip install matplotlib numpy

```

---

## 📖 9. References

1. Hillier & Lieberman, *Introduction to Operations Research* (2015)

2. Bazaraa, Jarvis, & Sherali, *Linear Programming and Network Flows* (2011)

---

## 🧑‍💻 Author

Developed by **Nafis**, passionate about mathematics, automation, and programming education.

For educational purposes only.

---