Modeling and Simulation

In the real world we have a problem that is described using the terminology relevant to the problem in a problem statement. From the real world, we can step into the abstract world that consists of the specification of the problem, valid assumptions. We can then create a conceptual model of the problem that can be mapped to mathematical model and that can be mapped to computational model that consists of computer science concepts, principles and method. The output from the abstract world can then be either predicted or interpreted to solve the real world problem.

Solving complex problems often involves multiple stages of abstraction and modeling. Let’s break down this process in detail with an example:

  1. Real World Problem: Let’s consider a logistics company that wants to find the most efficient route for delivering packages to various locations in a city, given the start and end points. This problem is based in the real world, using the company’s specific terminology and context.

  2. Abstract World - Problem Specification & Assumptions: In this stage, we generalize the problem, removing the specifics of the logistics company and instead talking about nodes (locations), edges (routes), weights (distances or times), a start point, and an end point. We might make certain assumptions at this point - for instance, we may assume that there are direct routes between all pairs of locations, or that the time to travel between two locations is always the same.

  3. Conceptual Model: Here, we conceptualize the problem in a way that we can start to apply mathematical or logical reasoning to it. In this case, we might model the problem as a graph, where each location is a node, and each route between locations is an edge. The problem is then to find the shortest path that visits all nodes.

  4. Mathematical Model: In the mathematical modeling phase, we describe the problem using mathematical structures and equations. In our example, this might involve setting up the problem as a Traveling Salesman Problem (TSP) or a Vehicle Routing Problem (VRP), well-studied optimization problems that have mathematical formulations.

  5. Computational Model: Here, we translate our mathematical model into a form that can be processed by a computer. For instance, we might implement the problem using an adjacency matrix to represent the graph and apply a heuristic or exact algorithm (like the Held-Karp algorithm for TSP) to find the most efficient route. This is where principles of computer science, such as algorithm design and data structures, come into play.

  6. Interpreting the Solution: Once we’ve found a solution in the abstract world, we need to translate it back into the real world. In our example, this would involve interpreting the solution of our algorithm (i.e., the order in which to visit the nodes) as a specific route for delivery in the city.

  7. Validation or Prediction: Finally, we might validate our solution by comparing it with historical data (if available) or use it to predict future outcomes (like the expected delivery times for the chosen route).

Each stage in this process involves different types of thinking and expertise, and the ability to move between these different stages is a key skill in problem-solving.

These seven steps can be described as the “Problem-Solving Process” or “Problem-Solving Cycle”. However, in the context of algorithmic problem solving, computer science, or computational thinking, this process is often referred to as “Abstraction” or “Modeling and Simulation”. Abstraction is the process of reducing complexity by ignoring irrelevant information to focus on crucial details, and modeling and simulation is the process of creating abstract representations (models) of real-world problems to simulate possible solutions.