What Is Mixed Integer Programming?

Mixed integer programming (MIP) is a mathematical optimization technique that solves problems involving a mix of continuous variables (which can have any value, including decimals and fractions), discrete variables (which must be countable whole numbers), and binary variables (which can only take values 0 or 1).

MIP is a powerful tool widely used in operations research and decision science to solve complex optimization problems. It can help in scenarios with constraints, limited sources, and discrete decisions.

For business decision-makers and data scientists, MIP provides a structured way to make the best decisions for real-world challenges, like resource allocation, production planning, workforce scheduling, and facility placements.

For developers, MIP models provide a systematic and structured approach to optimization, enabling the encoding of real-world constraints, objectives, and decisions into code. This removes ambiguity and ensures explainable AI. With scalability and robustness, MIP allows developers to solve large-scale, complex problems, with high precision and speed. Additionally, its modular design ensures that models are reusable and interpretable, making it easy to adapt solutions across various use cases—whether optimizing logistics today or workforce planning tomorrow. This approach empowers businesses to make data-driven, optimal decisions quickly and accurately.

By adopting MIP, businesses can:

• Ensure Optimal Decision-Making: MIP doesn’t just provide “a solution”—it guarantees the best solution possible within defined constraints, making decisions data-driven and efficient.
• Achieve Better Resource Utilization: From minimizing shipping costs to maximizing production throughput, MIP ensures resources like time, money, and manpower are allocated optimally, enabling businesses to do more with less.
• Evaluate Trade-Offs Systematically: MIP enables scenario analysis at scale, helping businesses evaluate critical trade-offs (cost versus speed, staff versus service levels) rather than relying on heuristics or gut instinct.

• Enhance Resilience Under Constraints: In a dynamic business environment, MIP models can be updated with new data, allowing businesses to adapt to disruptions (e.g., supply chain issues, labor shortages, or regulatory changes) and find the best possible solutions

When businesses use MIP, they can optimize their processes and increase profitability by making more informed, data-driven decisions faster and effectively. For developers, MIP offers a powerful framework for representing and solving complex decision problems with precision and structure. For businesses, it’s a strategic tool that transforms complexity into clarity, enabling optimized decisions under real-world constraints—whether it’s allocating budgets, designing workflows, or optimizing logistics. MIP ensures that every decision leads to the best possible outcome.

MIP is a two-step process: problem modeling and problem solving, which allow businesses and developers to optimize decision-making across various complex scenarios.

In the first step, users—such as developers, analysts, or data scientists—formulate the real-world problem by identifying key components and expressing them mathematically. This is known as problem modeling.

Key model components are:

• Decision Variables: These represent the choices to be made. They can be:
  • Binary: 0 or 1 decisions (e.g., whether to open facility or not)
  • Discrete: Countable values (e.g., number of items to produce)
  • Continuous: Any real value (e.g., amount of raw material used)
• Objective Function: This is the goal of the optimization. The objective could be:
  • Single-Objective:
    • Maximization: E.g., maximizing profit or revenue
    • Minimization: E.g., minimizing cost or time
  • Multi-Objective:
    • Prioritized: One objective is more important then others.
    • Combined: More objectives are combined into a single numeric objective.
    • Represented as Constraints: Objectives are framed as constraints on target goals.
• Constraints: These are the boundaries within which the decision variable must operate. Constraints can be:
  • Logical: E.g., total number of stored items can’t exceed warehouse capacity
  • Physical: E.g., material or space limitations
  • Capacity-Based: E.g., labor hours or machine capacity
  • Policy-Driven: E.g., company or regulatory rules

Figure 2. Example of a mixed integer programming (MIP) model and its solution space: x and y are the decision variables, and z is the objective function. The inequalities form the constraint boundaries, represented as lines. Blue dots indicate feasible (valid) solutions that satisfy all constraints, while the green dot marks an optimal solution that maximizes the objective.

Once the problem is modeled, it must be solved using algorithms that navigate through the solution space to find the optimal solution.

Here’s how MIP is typically solved:

1. Model Encoding/API Integration:
   After modeling the problem, the model is encoded using a supported modeling language or APIs. Common frameworks include AMPL, CVXPY, PuLP, Pyomo, and SciPy. The model is then integrated with a solver interface.
2. Solving the Model:
   The solver interacts with the encoded model either through standard formats (e.g., LP, MPS, NL) or APIs like solver.solve(model). The solver performs several steps.
   • Parsing the Model: Understanding the structure of the problem
   • Pre-Solving: Using techniques like linear relaxations, constraint tightening, and bound propagation to simplify the problem
   • Solving: Applying algorithms such as branch-and-bound, cutting planes, and heuristics to efficiently navigate the solution space and solve the model efficiently

   The solver’s output will include:

   • Optimal Decision Variables Values: E.g., the best values for production quantities or resource allocation
   • Optimal Objective Function Value: E.g., the best profit or minimum cost
   • Solver Status: E.g., optimal, infeasible, unbounded
3. Solution Interpretation:
   Once the solution is found, the user interprets the output by analyzing the results (e.g., determining the best routes to take or how much to produce). Additionally, the user may examine dual values, slacks, or solve multiple scenarios to assess different trade-offs and solutions.
4. Deployment and Automation:
   The MIP solution is often integrated into a larger decision system, such as supply chain planning, workforce allocation, or production scheduling. The solution can be deployed in various settings:
   • Batch Processes: Running optimization at scheduled intervals
   • Interactive Applications: Providing real-time optimization support for decision-makers
   • Real-Time Decision-Making Engines: Enabling automated responses based on constantly changing inputs (e.g., in logistics or emergency services)

To tackle real-world problems efficiently, various algorithms are employed to find the best solutions while managing constraints and resources.

One such technique is the linear programming (LP) relaxation of an MIP model, where the integrity requirement for integer variables is relaxed (i.e., removed). This provides an upper bound for a maximization problem or a lower bound for a minimization problem, which can then be used in combination with the branch-and-bound technique to guide the search for an optimal solution.

The table below outlines some of the most widely used algorithms in MIP, explaining the core principles, typical use cases, and their impacts on improving the efficiency of solving MIP problems.