Optimize with Gurobi Python: GurobiPy Example Code

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Linear programming (LP) serves as a powerful tool for tackling optimization challenges across various industries. In this guide, we delve into the world of LP using the gurobipy library, a Python interface for Gurobi, one of the most robust and speedy optimization solvers available. The company consistently introduces new features to enhance its capabilities. For those interested in the licensing details, they can be found on Gurobi’s website.

The significance of LP lies in its ability to address complex decision-making problems that involve allocating limited resources to maximize profits or minimize costs. Whether it’s in the domains of manufacturing, finance, logistics, or any other field, LP provides a structured approach to find optimal solutions;

The gurobipy library, with its seamless integration into Python, simplifies the process of modeling and solving LP problems, making it accessible to a wide range of professionals and researchers. Its efficiency, coupled with Python’s versatility, allows users to explore and experiment with different scenarios, gaining valuable insights into their operations.

As industries continue to evolve and face increasingly intricate challenges, LP remains a valuable ally for making data-driven decisions and achieving efficiency, cost savings, and competitive advantages. By mastering LP techniques with tools like gurobipy, individuals and organizations can navigate the complexities of modern problem-solving with confidence and precision.

Getting Started

To embark on this tutorial, you must have the gurobipy Python library installed. If you haven’t done so already, please follow these steps:

Open “Command Prompt” (on Windows);
Execute the following code:

“`

python -m pip install -i https://pypi.gurobi.com gurobipy

“`

Limited License Note

Gurobipy includes a limited license to facilitate library exploration and the solution of sample optimization problems, making it accessible for users to get acquainted with its capabilities and functionality. This initial access is invaluable for learning and experimentation, allowing individuals to gain hands-on experience.

However, for more complex and real-world optimization challenges, acquiring a full license becomes imperative. A licensed version of Gurobi provides access to advanced features, optimization algorithms, and the ability to solve larger, more intricate problems. This is crucial for businesses and organizations seeking to leverage the full potential of Gurobi in addressing their specific needs and maximizing efficiency.

While the limited license serves as an excellent starting point, acquiring a full license is a strategic investment for tackling complex optimization tasks and harnessing the true power of Gurobi in driving informed decision-making and achieving optimal results.

Linear Programming Example

Understanding linear programming becomes more intuitive when explored through a practical example. Imagine a manufacturing company that specializes in producing cups and plates. Here’s the essential information for this scenario:

Cups fetch a selling price of $27 each, while plates are sold for $21;
Manufacturing each cup incurs $10 in materials and $14 in labor costs, whereas plates cost $9 in materials and $10 in labor per unit;
Crafting a single cup demands 2.2 hours of labor, while a plate requires only 1 hour;
Raw materials are abundantly available, removing any constraints on material supply;
However, there is a limitation of 100 labor hours available due to the number of workers;
While there’s no limit on the demand for cups, the demand for plates stands at 30 units;
The ultimate goal of the company is profit maximization, calculated as revenue minus cost.

This practical example sets the stage for exploring linear programming as we aim to maximize profits by optimizing the production quantities of cups and plates, taking into account resource constraints and demand limitations. Through this example, we illustrate the step-by-step process of formulating and solving a linear programming problem, shedding light on the power of this technique in real-world decision-making scenarios.

How to Solve a Linear Programming Problem

Step 1: Define Decision Variables

Every linear programming problem necessitates the identification of decision variables, which serve as the core components representing the key choices within the problem. In our scenario, the pivotal decisions revolve around determining the quantities of cups and plates to produce. These decision variables are denoted as x1 for the number of cups to produce and x2 for the number of plates to manufacture.

Through these variables, the manufacturing company can precisely define its production strategy. By optimizing these values within the constraints of resources, labor, and demand, the company can achieve the primary objective of maximizing profits. These decision variables lay the foundation for constructing the objective function, which quantifies the company’s profit goals and drives the entire linear programming optimization process.

Step 2: Formulate the Objective Function

Optimization problems invariably aim to maximize or minimize something. In our case, the company seeks to maximize profits. To achieve this, we construct the objective function as follows:

Revenue = $27 * x1 + $21 * x2;
Raw Material Cost = $10 * x1 + $9 * x2;
Labor Cost = $14 * x1 + $10 * x2;
Profit = Revenue – Raw Material Cost – Labor Cost = $3 * x1 + $2 * x2.

Hence, our objective function is to maximize z = $3 * x1 + $2 * x2.

Step 3: Establish Constraints

1. Labor Hours Constraint:

– 2.2 * x1 + x2 ≤ 100

2. Demand Constraint for Plates:

– x2 ≤ 30

3. Non-Negativity Constraints for Decision Variables:

– x1 ≥ 0

– x2 ≥ 0

Solving the Linear Programming Problem Using Python

With our optimization problem defined, we can now employ the gurobipy library in Python to solve it. Here are the steps:

Import the gurobipy library;
Create a new empty model;
Add decision variables x1 and x2 to the model;
Set the objective function as a maximization problem;
Add the constraints;
Optimize the model.

“`python

from gurobipy import *

Create a new model

m = Model()

Create variables

x1 = m.addVar(name="x1")
x2 = m.addVar(name="x2")

Set objective function

m.setObjective(3*x1 + 2*x2 , GRB.MAXIMIZE)

Add constraints

m.addConstr(2.2*x1 + x2 <= 100, "c1")
m.addConstr(x2 <= 30, "c3")
m.addConstr(x1 >= 0, "c4")
m.addConstr(x2 >= 0, "c5")

Optimize model

m.optimize()

Print values for decision variables

for v in m.getVars():
    print(v.varName, v.x)

# Print maximized profit value
print('Maximized profit:', m.objVal)
```

In practice, the company cannot produce fractional quantities of cups or plates. To address this, you can explore solving integer programming problems with Python.

Conclusion

In this extensive guide, we have ventured into the world of linear programming (LP) using the powerful Gurobi Python interface, gurobipy. LP is a versatile tool that finds applications in a myriad of industries for optimizing decision-making processes. Through a practical example, we illustrated how to formulate and solve an LP problem step by step.

Beginning with the essential process of defining decision variables, we moved on to construct the objective function that encapsulates the primary goal of LP – in our case, profit maximization. We then established constraints that reflect real-world limitations, such as labor hours and demand constraints;

With the problem thoroughly defined, we harnessed the capabilities of Python and gurobipy to efficiently find the optimal solution. We outlined the code to create the model, add variables and constraints, set the objective function, and ultimately optimize the model;

One critical aspect we touched upon is the need for integer programming when dealing with discrete quantities, such as the number of cups and plates a company can produce. This guide serves as a foundation for tackling more complex LP problems and exploring advanced topics in optimization.

Linear programming, with the support of powerful tools like Gurobi and the simplicity of Python, empowers organizations to make informed decisions, maximize efficiency, and ultimately enhance their bottom line. As you delve deeper into the realm of optimization, remember that LP is a valuable ally in the pursuit of excellence in various fields.