Why Model If I Can’t Solve It?

Sometimes, students ask why they should include a mathematical model in their research if it cannot be solved to optimality for larger instances or real-world scenarios. They wonder if it would be sufficient to jump directly to a simulation or heuristic approach, and compare its performance against the company’s benchmark.

For example, suppose the problem is a dynamic variant of the Job Shop Scheduling Problem (JSSP), which is NP-hard, arising in a warehouse AS/RS system. In that case, students might think that simulating the process and testing different dispatching rules would be enough. After all, the problem at hand is complex and involves real-time decision making, which makes it impractical to find an optimal solution for larger instances.

However, this approach is often not sufficient for several reasons, and including a model in your research is highly recommended.

In the following sections, we will explore why including a model is beneficial, how it contributes to the rigor of your research, when simulation alone might suffice, and the potential drawbacks of skipping the modeling phase.

We will also provide an example of how to present a model, as well as a heuristic approach, in your research.

Reasons to Include a Model

The main reasons to include a model in your research, even if it cannot be solved for larger instances, are:

  1. Formalization, Clarity, and Communication. A mathematical model clearly conveys what you are doing to an expert audience. It formally lays out the assumptions, constraints, objective function, parameters, and variables explicitly, communicating the exact nature of the problem with no room for ambiguity. This transparency is crucial for peer review, replication, and validation by other researchers.
  2. Theoretical Contribution. Developing a model demonstrates the student’s ability to engage with and contribute to the theoretical foundations of the field, especially when addressing a novel variant. It shows that you understand the literature and can formulate the problem in a way that adds value to the academic community.
  3. Problem Understanding and Analysis. A model helps in understanding the problem’s structure, identifying key variables, constraints, and relationships. It provides a framework for analyzing the problem systematically, which is essential for developing effective solution methods.
  4. Step-by-Step Development. Presenting a model first, even if it’s only solvable for small instances, allows for a structured narrative in your research. You can demonstrate that while the model works for small instances, it becomes impractical for larger ones. This leads naturally to the introduction of your simulation as a practical solution.
  5. Building Confidence in the Approach. Successfully solving the model, even for small instances, builds confidence in your approach and methodology. It demonstrates that you have a solid grasp of the problem and the tools required to address it, which can be reassuring for both you and your examiners.
  6. Academic Rigor and Expectations. In many academic programs, particularly in Industrial Engineering and Management, there is an expectation that students will demonstrate their ability to formulate and solve mathematical models. Including a model aligns with these expectations and showcases your skills in mathematical modeling and optimization. MSc students are expected to have the skills to formulate and solve models. Therefore, including a model showcases that you are capable of fulfilling these expectations and are not avoiding the formalism due to a lack of understanding.

In summary, including a model will enhance the rigor of your research and demonstrate your understanding of the problem domain. Even if the final solution relies on simulation, the model adds significant value to the overall research process.

Reasons to Solve the Model

Sometimes, students include the model but do not solve it, which can lead to questions about the rigor and completeness of their research. Here are some reasons why solving the model, even for small instances, is beneficial:

  1. Proving Feasibility and Correctness. Solving the model confirms that your formulation is logically sound and practically usable. It validates that all constraints and objectives interact correctly, and it demonstrates the model is solvable and produces meaningful outcomes.
  2. Exploring Model Behavior and Robustness. By solving the model under different parameter settings, you gain insights into how it behaves. This allows for basic sensitivity analysis and helps identify critical parameters, increasing your understanding of the problem’s structure and robustness.
  3. Providing Benchmarks for Evaluation. Solutions to small instances act as ground truth for testing the quality of heuristics or simulations. You can assess optimality gaps, evaluate performance, and ensure that approximate methods perform reasonably well relative to known solutions.
  4. Informing Heuristic Development. Observing optimal solutions in small cases often reveals structural patterns or solution strategies that can be generalized. These insights help design effective heuristics, guiding decisions such as neighborhood structure or priority rules.
  5. Assessing Scalability and Complexity. Solving a range of small to medium instances helps you analyze how solution time and quality change with problem size. This illustrates computational complexity and justifies the need for heuristic or simulation approaches for larger instances.
  6. Laying a Foundation for Future Research. Documenting solved instances contributes to the academic literature and supports future benchmarking. This aligns your work with common practices in the field (e.g., VRP benchmarks such as the Solomon instances) and enables others to build upon or extend your results.

In summary, solving the model strengthens the credibility, depth, and academic value of your research. It validates your formulation, supports further analysis, and builds a foundation for the rest of your work. Even if the ultimate solution relies on heuristics or simulation, solving small instances of your model demonstrates rigor and prepares your research for comparison, publication, and reuse.

When A Model Might Not Be Necessary

While including a model is generally recommended, there are scenarios where jumping to simulation or heuristic approaches alone might be sufficient or even preferable.

  • Complex and Dynamic Systems: In contexts where the system is highly dynamic and complex with many interacting components and stochastic elements, simulations can often provide insights that are difficult to capture with static models.
  • Classical Problems with Established Models: If the problem is a well-studied classical problem (like JSSP) with established models and benchmarks, and your focus is on testing new heuristics or rules, it might be acceptable to skip the formal model. Notice, however, that it only applies if the problem is well-known. If you are dealing with a novel problem or a variant of a classical problem (new constraints, objectives, etc.), it is still advisable to include a model.

Hence, while there are exceptions, these cases are not the norm. The general expectation in academic research, particularly in fields like Industrial Engineering and Management, is that students will engage with the formalism of modeling to demonstrate their understanding and skills. While it is acceptable to skip the formal model, be aware that this approach might be viewed less favorably by examiners. They may see it as a less rigorous method unless you provide strong justification and demonstrate that you have the capability to treat the problem formally.

Example of Including a Model in Research

The following research article illustrates how to present a model and then a heuristic approach to solve a problem that is too large for the model to be solved in a reasonable time.

The article is titled “The Share-a-Ride Problem: People and parcels sharing taxis” by Li et al. (2014) (link). It presents a static Mixed Integer Linear Programming (MILP) model for the Share-a-Ride Problem (SARP) and then discusses a dynamic approach that is more practical for larger instances.

The SARP is a variant of the Dial-a-Ride Problem (DARP) where both people and parcels share taxis. In turn, the DARP is a variant of the Vehicle Routing Problem (VRP), a known NP-hard problem. Hence, it is clear that the SARP is also NP-hard, and solving it optimally for large instances is impractical. Still, the authors present a static MILP model to formalize the problem, show that it can’t be solved for large instances, and then propose a dynamic approach that is more feasible.

This way, they demonstrate the theoretical foundation of their work while also providing a practical solution for real-world applications.

The following steps outline how to present a model and then a heuristic approach in your research:

  1. Present the Static MILP and show the difference from typical models:

    “In this model, we considered the delivery of both people and parcels. Unlike the DARP that usually assumes that the amount of cars is large enough for serving all the requests (see, e.g., Cordeau & Laporte, 2003), the SARP allows rejections for both people and freight. If the people transportation network cannot serve all freight requests (e.g., the capacity is not large enough), some requests should be rejected and served by the freight network. Thus, the SARP implies cooperation between people and freight transportation networks. The traditional DARP model does not fulfil these requirements, which motivated us to build this SARP model.” (Li et al., 2014, p. 34)

  2. State that the problem is dynamic and that they will be compared.

    “Instead of static scenarios, we can envision a more realistic dynamic case in which passengers are accepted or declined at a time of their call (as usually happens), and then inserting feasible parcels within their route. Parcels are known beforehand and available for insertion dynamically. In particular, it would be useful to find out by experimentation whether the static and dynamic setting produce comparable results or indicate similar behavior.” (Li et al., 2014, p. 34)

  3. Present the dynamic approach.

  4. Solve the static MILP.

    “We started the numerical experiments with small instances, however, it turned out that these are already computationally demanding. That is why our experiments with the SARP are limited to only 12 small scenarios.” (Li et al., 2014, p. 36)

  5. Show that the static MILP can’t be solved for large instances in a reasonable time.

    “Table 3 shows the results for the static SARP model. Each number in the table is obtained by averaging over 10 instances, the corresponding models run within a time limit of 2 h. From the table, we see that by fixing the number of taxis, the profit increases as the number of requests grows. More computationally efficient algorithms are needed to solve the SARP model for realistic instances.” (Li et al., 2014, p. 37)

  6. Show the results of the dynamic approach.

Conclusion

In conclusion, including a model in your research is highly beneficial for clarity, theoretical contribution, methodological rigor, and benchmarking. It provides a solid foundation for your work, enhances the credibility of your findings, and demonstrates your capability as a researcher. While there are scenarios where jumping directly to simulation or heuristic approaches might be sufficient, these cases are exceptions rather than the rule. The general expectation in academic research, particularly in fields like Industrial Engineering and Management, is that students will engage with the formalism of modeling to demonstrate their understanding and skills.

Ultimately, the decision to include a model should be based on your timeline and grading expectations. If you have the time and resources, including a model can enhance the rigor and credibility of your work. Examiners will appreciate the effort and thoroughness involved in modeling the problem.