How can you identify and resolve infeasibility in linear programming problems?

1 Why is infeasibility a problem?

Infeasibility is a problem because it prevents you from finding the optimal solution to your problem. It also indicates that there is something wrong with your model or your data. For example, you may have made a mistake in defining your variables, parameters, objective function, or constraints. Or you may have used unrealistic or inconsistent data that does not reflect the actual situation. Infeasibility can also arise from logical errors, such as imposing contradictory or redundant constraints. In any case, infeasibility means that your model does not capture the essence of your problem and needs to be revised.

3 How to resolve infeasibility?

In order to resolve infeasibility, depending on the nature and source of the problem, there are several common methods that can be used. Relaxing or removing some constraints that are not essential or realistic can increase the feasible region. Additionally, adding or modifying variables that can better represent the problem can be beneficial. It is also important to check and correct data to ensure it is valid and reliable. Lastly, reformulating the model to capture the essence and structure of the problem can help. For example, changing the objective function or constraints to reflect goals and assumptions, or using a different mathematical formulation or technique can be useful.

4 How to verify feasibility?

After you have resolved infeasibility, you need to verify that your problem is indeed feasible and optimal. You can use a solver again to solve your modified problem and check the results. If the solver finds a feasible solution, it will report the optimal value of the objective function and the optimal values of the variables. You can also check the values of the slack or surplus variables to see if any constraints are binding or not. You can then analyze and interpret the results to see if they make sense and meet your expectations. If not, you may need to revise your model or data again.

5 How to prevent infeasibility?

To avoid infeasibility, it is important to follow best practices when modeling and solving linear programming problems. Firstly, you should understand your problem by defining variables, parameters, objective function, and constraints. Additionally, you need to use realistic and consistent data from reliable sources and appropriate units. Moreover, you should test your model for feasibility and optimality with a solver, perform sensitivity analysis and scenario analysis. Lastly, after solving the model, review and refine it for accuracy and relevance by analyzing the results and comparing them with your goals. Don't forget to update your model as new information or feedback becomes available.