The question is wrong: You can't prioritize the constraints. They should be all satisfied. Prioritization can only happen over the objective functions not constraints.

Linear programming with limited resources is analogous to having to prioritize limited bandwidth of a team to achieve maximum business impact. While it’s important to maximize business impact, one shouldn’t loose oversight of business constraints such as current commitments, long term vs. short term trade offs, individual specializations etc. Once you get the optimal plan of action, then it’s the time to implement. Or you can run additional possible scenarios through sensitivity analysis to see what if i had little bit more budget, how much more impact I can do. Good luck linear programming in your day to day work!

Constraints aren't equal in impact, hence strategic prioritisation is essential. Start by mapping your constraints to objective function coefficients, analysing their dual values for insights into binding conditions. Focus on shadow prices to understand the cost of tightening resources, identifying areas of high sensitivity. Resource allocation might also require blending methods like branch-and-bound for integer considerations, especially in complex supply networks. Real-world shifts, such as fluctuating lead times or demand surges, underscore the necessity for adaptive strategies. Sensitivity analysis and scenario modeling are critical to staying agile and data-driven.

Determine the binding constraints in the model by introducing slack variables into the defined constraints. A positive slack indicates that the constraint has some flexibility, while a slack variable with a value of zero signifies that the constraint is binding. One can also use dual values to assess the relative importance of these binding constraints.

I recently worked on a complex project with different level of constraints priorities. I used slack variables with some constraint where from business perspective we had some rooms to move. I then added the slack variables with weights as penalty in objective function. The approach allowed us to successfully find feasible and optimum solution to a very large and complex problem.