- A linear programming problem will consist of decision variables, an objective function, constraints, and non-negative restrictions.
- The decision variables, x, and y, decide the output of the LP problem and represent the final solution. The objective function, Z, is the linear function that needs to be optimized (maximized or minimized) to get the solution.
- The constraints are the restrictions that are imposed on the decision variables to limit their value. The decision variables must always have a non-negative value which is given by the non-negative restrictions.
- The general formula of a linear programming problem is given below:
- Objective Function: Z = ax + by
- Constraints: cx + dy ≤ e, fx + gy ≤ h.
- The inequalities can also be "≥" Non-negative restrictions: x ≥ 0, y ≥ 0 How to Solve Linear Programming Problems?
- The most important part of solving linear programming problem is to first formulate the problem using the given data. The steps to solve linear programming problems are given below:
- Step 1: Identify the decision variables.
- Step 2: Formulate the objective function. Check whether the function needs to be minimized or maximized. Step 3: Write down the constraints.
- Step 4: Ensure that the decision variables are greater than or equal to 0. (Non-negative restraint)
- Step 5: Solve the linear programming problem using either the simplex or graphical method.
- Linear Programming by Simplex Method The simplex method in lpp can be applied to problems with two or more decision variables.
- Step 1: Add another variable, known as the slack variable, to convert the inequalities into equations. Also, rewrite the objective function as an equation.
- Step 2: Construct the initial simplex matrix.
- Step 3: Identify the column with the highest negative entry. This is called the pivot column.
- Step 4: Divide the entries in the rightmost column by the entries in the pivot column. We exclude the entries in the bottom-most row.
- Step 5: With the help of the pivot element perform pivoting, using matrix properties, to make all other entries in the pivot column 0.
- Step 6: Check if the bottom-most row has negative entries. If no, then the optimal solution has been determined. If yes, then go back to step 3 and repeat the process. -10 is a negative entry in the matrix thus, the process needs to be repeated. We get the following matrix.
- Linear Programming by Graphical Method If there are two decision variables in a linear programming problem then the graphical method can be used to solve such a problem easily.
- Step 1: Write all inequality constraints in the form of equations.
- Step 3: Identify the feasible region. The feasible region can be defined as the area that is bounded by a set of coordinates that can satisfy some particular system of inequalities.
- Determine the coordinates of the corner points. The corner points are the vertices of the feasible region.
WHAT IS LINEAR PROGRAMMING?
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