Formulate a dual linear programming problem.

What is a dual linear programming problem?

A dual linear programming problem is a mathematical optimization problem that is derived from a given primal linear programming problem. The dual problem is obtained by interchanging the roles of the objective function and the constraints of the primal problem.

The dual problem is formulated as follows:

Maximize:
z = ∑(i=1 to n) bi*yi

Subject to:
∑(i=1 to n) ai,j*yi ≤ cj for j = 1,2,...,m
yi ≥ 0 for i = 1,2,...,n

where bi and cj are the coefficients of the primal problem, and ai,j are the coefficients of the constraints. The variables yi are the dual variables, which correspond to the constraints of the primal problem.

The dual problem has the same number of variables as the primal problem, but the number of constraints is equal to the number of variables in the primal problem. The objective function of the dual problem represents the minimum value of the primal problem, and the constraints of the dual problem represent the maximum values of the primal problem.

The dual problem is useful in solving the primal problem, as it provides information about the sensitivity of the primal problem to changes in the coefficients of the constraints. It also provides a lower bound on the optimal value of the primal problem.

In conclusion, a dual linear programming problem is a mathematical optimization problem that is derived from a given primal linear programming problem by interchanging the roles of the objective function and the constraints. It is useful in solving the primal problem and provides information about its sensitivity to changes in the coefficients of the constraints.