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Dive deep into sample models, built with our Python API.
In this example, you’ll discover how mathematical optimization can be used to address a macroeconomic planning problem that a country may face. We’ll show you how to model and solve an input-output problem encompassing the interrelationships between the different sectors of a country’s economy. This model is example 9 from the fifth edition of Model Building in Mathematical Programming by H. Paul Williams on pages 263-264 and 316-317. This modeling example is at the intermediate level, where we assume that you know Python and are familiar with the Gurobi Python API. In addition, you should have some knowledge about building mathematical optimization models.
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Mathematical optimization uses the power of math to find the best possible solution to a complex, real-life problem. It can be thought of as a way to make the smartest (and most optimal) decision despite having a multitude of variables and challenges.
Mathematical optimization models contain three components:
1. Objective Function: This is the end goal that you want to achieve.
2. Decision Variables: These represent the items involved that you can control and change in order to reach your objective.
3. Constraints: These are the rules and/or limitations that you must follow.
To help put the idea of mathematical optimization into perspective, imagine that you’re a delivery company that’s trying to minimize the amount of time it takes to complete your delivery route.
Objective Function
Your objective function is simply to minimize the amount of time it takes to complete your delivery route. This might be driven by goals such as minimizing vehicle usage to reduce carbon emissions or saving on labor and fuel expenses.
Decision Variables
A delivery company can adjust key aspects of its operations to achieve its objective. For instance, changing the route taken could reduce travel distance and time, while altering the time of day to start deliveries could help avoid peak traffic, ensuring faster and smoother operations.
Constraints
The constraints that you are limited by can range from the number of deliveries that you’re required to make, the times that you’re required to deliver by and the locations you’re delivering to. These factors cannot be changed as they are a core part of the business’ services.
Descriptive and predictive analytics show you what has happened in the past, why it happened, and what’s likely to happen next. But to decide what to do with that information, you need human input—which can introduce bias.
With mathematical optimization, you receive a decision recommendation based on your goals, constraints, and variables alone. You can, of course, involve human input when it comes to whether or not to act on that recommendation. Or you can bypass human input altogether and automate your decision-making.
80% of the world’s leading companies use mathematical optimization to make optimal business decisions. It provides stakeholders with decisions that are data-driven and free from subjective human bias. Mathematical optimization has countless real-life uses, such as managing supply chains to minimize costs, planning production schedules in manufacturing, and improving staff scheduling to improve efficiency.
For example, Air France uses it to build the most efficient schedule for its entire fleet to save on fuel and operational costs whilst reducing delay propagation. You can read more about this here.
Regardless of the industry that you work within, if you have a complex problem that needs resolving, Gurobi Optimization could be the solution that you’ve been seeking.
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