Chapter 2 II SIMPLEX METHOD OF SOLVING LPP.pptx
- 1. SIMPLEX METHOD OF SOLVING LPP
- 2. Simplex Algorithm Method The graphical method to solving LPPs provides fundamental concepts for fully understanding the LP process. However, the graphical method can handle problems involving only two decision variables (say X1 and X2). In 19940’s George B. Dantzig developed an algebraic approach called the Simplex Method which is an efficient approach to solve applied problems containing numerous constraints and involving many variables that cannot be solved by the graphical method.
- 3. Simplex Algorithm Method... The simplex method is an ITERATIVE or “step by step” method or repetitive algebraic approach that moves automatically from one basic feasible solution to another basic feasible solution improving the situation each time until the optimal solution is reached at.
- 4. Simplex Algorithm Method... Note: The simplex method starts with a corner that is in the solution space or feasible region and moves to another corner. The solution space improving the value of the objective function each time until optimal solution is reached at the optimal corner.
- 5. Simplex Algorithm Method... Scope of solution of LPP by simplex method Following types of problems are solved by simplex method: Maximize Z with inequalities of constraints in “< “form. Minimize Z with inequalities of constraints in “>“form. Maximize Z or Minimize Z with inequalities of constraints in “< “, “>“or “=”form.
- 6. certain basic terms relevant for solving LPP through simplex procedure Standard form: a linear programming in which all the constraints are written as equalities. Example: 2X1+3X2≤3 written as 2X1+3X2=3. Slack variable: a variable added to the left hand side of a ‘less than ‘ or ‘equal to’ constraint to convert the constraint to an equality is called slack variable. Example: 2X1+3X2≤3 converted as 2X1+3X2 +s1=3 In economic terminology, slack variable represents unused resources (capacity).
- 7. certain basic terms relevant for solving LPP through simplex procedure... Surplus variable: a variable subtracted from the left hand side of the ‘greater than or equal to’ constraint, to convert the constraint in to an equality is called surplus variable. Example 2X1+3X2≥3 converted as 2X1+3X2 -s1=3
- 8. certain basic terms relevant for solving LPP through simplex procedure... Optimal solution: any basic feasible solution which optimizes the objective function of a general LP problem is called an optimal basic solution. Simplex tableau: a table used to keep track of the calculations made of each iteration when the simplex solution is employed. Z; row: the number in this row under each variable represents the total contribution of outgoing profit when one unit of a non-basic variable is introduced in to the basis in place of a basic variable.
- 9. certain basic terms relevant for solving LPP through simplex procedure... Cj-Zj row: the row containing the net profit (loss) that will result from introducing one unit of the variable indicated in that column in the solution. Pivot (key) column: the column with the largest positive number in Cj-Zj row of maximizationminimization problem. It indicates which variable will enter the solution next.
- 10. certain basic terms relevant for solving LPP through simplex procedure... Pivot (key) row: the row corresponding to the variable that will leave the basis in order to make place (room) for the entering variables. The departing variable will correspond to the smallest positive ratio found by dividing the quantity column values by the pivot column values for each row. Pivot (key) element/number: the element at the intersection of pivot row and pivot column.
- 11. Steps to simplex Problem 1. Standardize the problem by introducing the slack and surplus in both the objective function and constraint 2. Draw the initial simplex tableau 3. Select the column with highest Cj-Zj as pivot/key column having entering variable 4. Perform the replacement ratio/minimum ratio 5. Choose the raw with minimum solution quantity as a pivot/key raw having leaving/outgoing variable to be replaced by the entering variable in the pivot/key column 6. Divide each element of the pivot row by the pivot element to find new values in the key or pivot row. 7. Perform row operations to make all other entries for the pivot column equal to zero