Why model if I can’t solve it?

Reasons to include a model

Include a model for the following reasons, even when large instances remain intractable:

1. Formalization, clarity, and communication. A mathematical model states the assumptions, constraints, objective function, parameters, and variables explicitly. Expert readers can then inspect, replicate, and validate the problem definition.
2. Theoretical contribution. A model shows how a new variant relates to established formulations. It demonstrates your understanding of the literature and identifies the formal change contributed by the thesis.
3. Problem understanding and analysis. Formulation forces you to identify the key variables, constraints, and relationships. That structure informs the design of an effective solution method.
4. Step-by-step development. Solve the model on small instances and document where it becomes impractical. The computational limit then motivates the simulation or heuristic used for larger instances.
5. Confidence in the approach. A solved small instance verifies that the formulation produces meaningful outcomes. It also shows you and the examiners that you understand the problem and the tools used to address it.
6. Academic rigor and expectations. Industrial Engineering and Management programs commonly expect MSc students to formulate and solve mathematical models. The model demonstrates those skills and makes the formal reasoning available for assessment.

A model strengthens the rigor of the research and demonstrates your understanding of the problem domain. It remains useful when the final solution relies on simulation.

Reasons to solve the model

An unsolved model leaves its correctness and usefulness uncertain. Solve at least small instances for the following reasons:

1. Feasibility and correctness. A solution confirms that the formulation is logically consistent and usable. It tests whether the constraints and objective interact as intended.
2. Model behavior and sensitivity. Solutions under different parameter settings show how the model behaves and which parameters affect the outcome most.
3. Benchmarks for evaluation. Optimal solutions to small instances provide reference values for heuristics and simulations. Use them to calculate optimality gaps and assess approximate methods.
4. Heuristic development. Optimal solutions can reveal structural patterns that inform neighborhood design, priority rules, and other heuristic choices.
5. Scalability and complexity. Solving small to medium instances shows how runtime and solution quality change with problem size. The results justify approximate methods for larger instances.
6. Future research. Published solutions support later benchmarking and extension. This follows established practice in fields such as vehicle routing, which uses benchmark sets such as the Solomon instances.

Solving the model validates the formulation and supports the remaining analysis. Small solved instances prepare the work for comparison, publication, and reuse even when heuristics or simulation handle the final problem size.

When a new model may be unnecessary

A simulation or heuristic may be sufficient in the following situations:

• Complex dynamic systems: Simulation may capture interactions and stochastic behavior that a static model omits.
• Classical problems with established models: A well-studied problem such as JSSP may already have accepted formulations and benchmarks. A thesis focused on new heuristics or dispatching rules can cite an established model. A new variant with different constraints or objectives still requires a precise formulation.

Industrial Engineering and Management examiners generally expect students to engage with formal modeling. Cite the established formulation and justify your decision when the thesis does not present a new model. The justification must show that you can treat the problem formally and that the chosen approach fits the research question.

Example of a model followed by a heuristic

The following article presents a model and then a heuristic for instances that the model cannot solve within 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. The SARP is therefore also NP-hard, which makes optimal solutions impractical for large instances. The authors use a static MILP to formalize the problem and establish its computational limits. They then propose a dynamic approach for larger instances.

The combination documents the theoretical basis of the work and provides 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 how it differs 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

Include a model when the thesis introduces a new problem or variant. The formulation clarifies the assumptions, supports theoretical analysis, and provides benchmarks for approximate methods. Simulation or heuristics alone require a clear justification and an established formal problem definition.

Base the decision on the research question, timeline, and grading expectations. When time and resources allow, formulating and solving the model gives examiners direct evidence of your mathematical reasoning and care.