Algebraic Method of Linear Programming

The algebraic method of linear programming is useful for problems with two decision variables and two constraints. It helps find optimal decision variables and maximum profit, unlike graphical methods that can be more complex.

The four steps are: 1. Change constraint inequalities to equalities. 2. Eliminate one unknown variable and solve for the second. 3. Plug the solution into either equation to solve for the first unknown. 4. Plug both decision variables into the objective function to find the maximum profit.

The first example, Clickman Electronics, demonstrates the method. The objective function is to maximize Z = 7x1 + 5x2, subject to two constraints: 4x1 + 3x2 ≤ 240 and 2x1 + 1x2 ≤ 100. The inequalities are converted to equalities, and one variable is eliminated by multiplying one equation by -2. This leads to x2 = 40. Plugging this back into an equation yields x1 = 30. Finally, plugging x1 and x2 into the objective function gives a maximum profit (Z) of 410.

The second example maximizes Z = 4x1 + 3x2, with constraints 6x1 + 4x2 ≤ 48 (material) and 4x1 + 8x2 ≤ 80 (labor). Following the same steps, the inequalities become equalities. Multiplying the first equation by -2 eliminates a variable, resulting in x1 = 2. Plugging x1 = 2 into an equation yields x2 = 9. Substituting these values into the objective function gives a maximum Z of 35.

The algebraic method is a quick and efficient way to solve linear programming problems with two decision variables and two constraints, providing a direct solution for optimal variables and maximum profit without needing complex software like Microsoft Excel Solver.