Formulate a linear programming problem.

What is a linear programming problem?

A linear programming problem is a mathematical optimization problem where the objective function and constraints are linear. The goal is to maximize or minimize the objective function subject to the constraints.

The general form of a linear programming problem is:

Maximize or Minimize Z = c1x1 + c2x2 + ... + cnxn

Subject to:

a11x1 + a12x2 + ... + a1nxn ≤ b1
a21x1 + a22x2 + ... + a2nxn ≤ b2
...
am1x1 + am2x2 + ... + amnxn ≤ bm
x1, x2, ..., xn ≥ 0

where Z is the objective function to be maximized or minimized, c1, c2, ..., cn are the coefficients of the variables x1, x2, ..., xn in the objective function, aij are the coefficients of the variables x1, x2, ..., xn in the ith constraint, bi is the right-hand side of the ith constraint, and x1, x2, ..., xn are the decision variables.

The constraints represent the limitations or restrictions on the decision variables, and the non-negativity constraints ensure that the decision variables cannot take negative values.

Linear programming problems can be solved using graphical methods or algebraic methods such as the simplex method. The optimal solution is the set of values of the decision variables that maximize or minimize the objective function while satisfying all the constraints.