when doing minimization problems and there is more than one negative constant in the far right column is there a method for choosing the proper one to go with or is the process entirely arbitraru

When solving minimization problems using linear programming, the goal is to find the values of decision variables that minimize the objective function, subject to certain constraints. The decision variables can take on any values within certain bounds, which are usually represented by inequalities.

To determine which negative constant to choose in the far right column, you need to examine the coefficients in the objective function and the constraints. The key is to identify the negative constant that corresponds to a constraint that is binding or active at the optimal solution. A binding constraint is one that is satisfied as an equality (i.e., it holds with equality), and it is often indicated by a zero slack variable.

Here is an approach to help you choose the proper negative constant:

1. Solve the linear programming problem using a method like the simplex algorithm or graphical method.
2. Once you have the optimal solution, examine the slack variables associated with each constraint. A slack variable measures how much the left-hand side of a constraint exceeds its right-hand side.
3. If you find a zero slack variable for a particular constraint, it means that this constraint is binding at the optimal solution. In other words, the resources are fully utilized or exhausted for that constraint.
4. Look at the corresponding negative constant in the far right column of that particular constraint. This negative constant represents the contribution of this constraint to the objective function in the form of a cost or penalty.
5. Choose the negative constant that corresponds to the binding constraint as the one to minimize in the objective function.

By selecting the negative constant associated with the binding constraint, you are effectively minimizing the objective function while respecting the constraints that are most constrictive.

Remember, this approach assumes linearity and convexity in the problem and feasibility of the model. In more complex scenarios, such as non-linear programming or mixed-integer programming, different methods and techniques may be needed to optimize the objective function.