Optimization Models 101: Types, Tips, and Resources

Mathematical optimization equips business leaders with the information they need to navigate complex landscapes, make nuanced decisions, and work within a myriad of constraints. 

If your company is interested in incorporating mathematical optimization into their operations, one of the first things you may want to know is how an optimization model works. 

What Is an Optimization Model? 

An optimization model is a framework that mathematically represents your problem and is used to determine the best possible outcome—such as maximizing profits or minimizing costs—given a set of quantifiable objectives and constraints. 

While each solution is unique, the workflow for developing an optimization model is fairly standard: 

1. Develop a Problem Statement: Before you can solve a problem, you must first define it. A good problem statement should identify the particular metric of interest and whether the optimization model should maximize or minimize it.
2. Formulate the Model: Once you have defined the problem, you must identify the variables (i.e., the decisions you can make and control), how they come together to form the constraints that affect the outcome, and as well as the model’s objectives. Then, translate those components into a mathematical representation that can be read by a solver.
3. Leverage a Solver: After coding the problem, you can then leverage a mathematical optimization solver like Gurobi to identify the optimal solution quickly and reliably. 

Three Components of Every Optimization Model

While they range in complexity, every optimization model has three components: 

• The Objective Function: This is the metric that you seek to optimize. “Optimization” means maximizing or minimizing the objective function. For example, a company may want to maximize profits or minimize greenhouse gas emissions. 

• Decision Variables: These are the factors that impact the objective function if they change. Decision variables could include materials, equipment, labor hours, or financial investment.  

• Constraints: These are the limits or conditions that must be met, no matter what. For example, if a particular production line has five team members assigned to it, one constraint might be that labor cannot exceed 200 hours per week.

Types of Optimization Models  

The main types of optimization models are: 

• Linear Programming: These optimization models are appropriate for when all of the decision variables have a linear relationship with the objective function and constraints, and the variables are continuous (i.e., non-integers). For example, the Emergency Supply Pre-Positioning Strategy (ESUPS) used linear programming to determine where to allocate life-saving humanitarian supplies ahead of crises so that first responders have the supplies they need, where they need it, and at the right time. 

• Mixed-Integer Programming: Mixed-integer programming can accommodate both continuous and integer variables. To illustrate, the Norwegian national postal service used mixed-integer programming to minimize travel distance for postal deliveries, balance workloads across facilities, and account for terminals capacity and limitations. 

• Nonlinear Programming: Nonlinear models can optimize functions in scenarios where some variables have nonlinear impacts on the objective function. An example of nonlinear relationships is grocery pricing and profits. As prices increase, demand often decreases. This dynamic affects profits in nonlinear ways. Price strategy firm Luca uses nonlinear programming to advise grocery stores on the optimal price for thousands of products. This allows Luca’s customers to maximize profits and adjust to changing conditions in real time. 

• Quadratic Programming: Quadratic programming is used when the objective function includes quadratic terms, and constraints are linear. It’s often applied in finance (e.g., portfolio optimization), energy systems, and certain machine learning models. In quadratically constrained programming, both the objective function and the constraints are quadratic functions. SwissQuant used quadratic programming and mixed-integer quadratic programming to develop optimal investment strategies that are tailored to each customer’s profile. This approach achieved 1% higher returns for customers. 

How to Get Started with Optimization Modeling 

Ready to learn more about optimization modeling? Gurobi makes it easy to get started with a large collection of learning resources. Beginners may be particularly interested in: 

• Gurobi AI Modeling is Gurobi’s set of resources designed to help you leverage generative artificial intelligence (AI) for mathematical optimization. This enables learners to begin developing optimization models quickly, while avoiding common pitfalls. 

• OptiMods is Gurobi’s open-source Python repository of implemented optimization use cases using Gurobi. In addition to reading the explanations on how the models were designed, learners can apply the mods to their real life cases. 

• Jupyter notebooks: Learn how to build mathematical optimization models of real-world business, engineering, or scientific problems using Python. 

Visit gurobi.com/learn to access these and other optimization resources, including games, trainings, and more.