Learn how to develop an effective integer optimization model for your operations research problems. Find out how to define, formulate, solve, analyze, validate, and refine your model.

Defining parameters such as the "MIP Gap" (the gap between the theoretically best possible solution and the solution found by a solver) at the beginning is crucial for efficient optimization. Aiming for a solution that is too tight, such as a 0.001% optimal solution, is often not necessary and makes no sense anyway due to inaccuracies in the representation and modeling of the business problem. Adjusting the MIP gap to a more realistic value, such as 3%, can dramatically speed up the solution process and is still a good solution that is practically relevant for most applications. This approach allows for faster evaluation of solutions, making it an effective strategy for handling complex integer optimization problems.

How can you develop an effective integer optimization model?

Integer optimization models are mathematical tools that help you find the best solutions for problems that involve discrete variables, such as assigning tasks, scheduling activities, or routing vehicles. However, developing an effective integer optimization model can be challenging, as you need to balance the complexity, accuracy, and feasibility of your model. In this article, you will learn some tips and techniques to design and implement an integer optimization model that meets your objectives and constraints.

1 Define the problem

The first step to develop an effective integer optimization model is to clearly define the problem you want to solve. You need to identify the decision variables, the objective function, and the constraints that characterize your problem. The decision variables are the unknowns that you want to determine, such as the number of units to produce, the order of tasks to perform, or the routes to follow. The objective function is the measure of performance that you want to optimize, such as the profit, the cost, or the time. The constraints are the limitations or requirements that you need to satisfy, such as the capacity, the demand, or the precedence. You should also specify the type and range of the decision variables, such as binary, integer, or mixed-integer.

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2 Formulate the model

The next step is to formulate the model using mathematical expressions that represent the objective function and the constraints in terms of the decision variables. You should use consistent notation and symbols to define your model, and follow the conventions and standards of your field or software. You should also check the validity and consistency of your model by verifying that it reflects the logic and assumptions of your problem, and that it does not have any errors or contradictions. You can use simple examples, diagrams, or tables to test and illustrate your model.

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Mark Zais, PhD

Data Science & Analytics Expert | Senior Technical Advisor | Principal Investigator | Operations Research Expert
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With complex problems, I find it best to work in three (iterative) phases: (1) mathematical equations, (2) convert to matrix format, (3) convert to programming language. Consider a complex multi-knapsack problem that is Strongly NP-Complete, unary NP-hard. First I whiteboard the objective function and constraints (this takes the most time). Then, I convert to an integer program in matrix format (Ax<=b). Lastly, using R (or software of choice), I write functions to load data, build the matrix, and solve the problem as a mixed integer programming problem with a library such as lpsolve. (Sorry! I realized that the word programming has multiple meanings here.)

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Dhruvi Goyal

Data Scientist | Researcher | Author | Dual University Gold Medalist
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In addition to this, a good strategy is also to check that the formulation is efficient, which means that we use less number of variables & constraints to represent our problem, and eliminate any redundancy from it. An optimized model formulation generally takes lesser solver's time as compared to an inefficient formulation.

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Vasiliy Zamolotov

Manager (ex-SAS, ex-Accenture) | Operations Research | Data Science Lead
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To create an effective model with integer variables, in addition to forming a mathematical model based on a business formulation, it is very important to have an idea of the algebraic form of the objective function and constraints. It is also important to understand how restrictions are related to each other, whether there is symmetry, how linear relaxation works when applying branch and bound algorithms. To increase the efficiency of the solution and reduce time, it is important to consider the problem through the prism of all these concepts and carry out the work. This is the only way to achieve an excellent model.

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3 Solve the model

The third step is to solve the model using an appropriate method or software that can handle integer optimization problems. Integer optimization problems are generally harder to solve than continuous optimization problems, as they require finding the optimal solution among a finite but large set of possible solutions. Therefore, you need to choose a method or software that can efficiently explore and evaluate the solution space, and that can handle the specific features and challenges of your problem, such as nonlinearity, multimodality, or uncertainty. You should also set the parameters and options of your method or software according to your preferences and needs, such as the accuracy, the speed, or the robustness of the solution.

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Dirk Zechiel

Founder & CEO @ Quantagonia | Gurobi GmbH founder | DecisionAI
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Defining parameters such as the "MIP Gap" (the gap between the theoretically best possible solution and the solution found by a solver) at the beginning is crucial for efficient optimization. Aiming for a solution that is too tight, such as a 0.001% optimal solution, is often not necessary and makes no sense anyway due to inaccuracies in the representation and modeling of the business problem. Adjusting the MIP gap to a more realistic value, such as 3%, can dramatically speed up the solution process and is still a good solution that is practically relevant for most applications. This approach allows for faster evaluation of solutions, making it an effective strategy for handling complex integer optimization problems.

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4 Analyze the results

The fourth step is to analyze the results of your model and evaluate their quality and relevance. You need to check if the solution is feasible, optimal, and unique, and if it meets your expectations and objectives. You should also examine the sensitivity and stability of the solution, and how it changes with respect to variations in the inputs, parameters, or assumptions of your model. You can use graphical or numerical tools to display and compare the results, and to identify the trade-offs and interactions among the decision variables, the objective function, and the constraints.

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5 Validate the model

The fifth step is to validate the model and assess its reliability and applicability. You need to compare the results of your model with the actual or expected outcomes of your problem, and measure the accuracy and error of your model. You should also test the model with different scenarios, data sets, or situations, and observe how it performs and adapts to different conditions and environments. You can use historical, experimental, or simulated data to validate your model, and use statistical or logical methods to measure and improve its fit and performance.

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Sreenath S

Data Science | Optimization | Machine Learning | IIT Bombay
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Validation of model and benchmarking it using standard instances is an important step because integer models are mostly about making decisions. Validation of model makes sure that there is no missing link and it can further be used redefine the model if needed.

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6 Refine the model

The final step is to refine the model and enhance its effectiveness and efficiency. You need to identify and address the limitations, weaknesses, or gaps of your model, and look for ways to improve its structure, formulation, or solution. You should also consider the feedback, suggestions, or criticisms from your stakeholders, clients, or peers, and incorporate their perspectives and needs into your model. You can use various techniques to refine your model, such as simplifying, reformulating, or adding variables, constraints, or objectives, or using different methods, software, or parameters to solve your model.

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7 Hereâ€™s what else to consider

This is a space to share examples, stories, or insights that donâ€™t fit into any of the previous sections. What else would you like to add?

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Directory

How can you develop an effective integer optimization model?

1

2

3

4

5

6

7

1 Define the problem

2 Formulate the model

3 Solve the model

4 Analyze the results

5 Validate the model

6 Refine the model

7 Hereâ€™s what else to consider

Operations Research

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Directory

How can you develop an effective integer optimization model?

1

2

3

4

5

6

7

1 Define the problem

2 Formulate the model

3 Solve the model

4 Analyze the results

5 Validate the model

6 Refine the model

7 Hereâ€™s what else to consider

Operations Research

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