Solve The Linear Programming Problem By The Simplex Method

The above information can be expressed in the Table 1 In the table, first row represents coefficients of objective function, second row represents different variables (first regular variables then slack/surplus variables).

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Infinite Number of Solutions: A Linear Programming Problem is said to have infinite number of solutions if during any iteration, in Cj-Ej row, we have all the values either zero or -ve. But since one of the regular variables has zero value in Cj-Ej row, it can be concluded that there exists an alternative optimal solution. It can be seen that optimal solution has been reached since all values in Cj-Ej row are zero or -ve But X is non basic variable and it has zero value in Cj-Ej row, it indicates that X, can be brought into solution, however it will not increase the value of objective function and alternative optimal exists. Maximize Z = Y, 2Y are basic variables (variables in current solution)to start with. AS Cj- Ej is positive, the current solution is not optimal and hence better solution exists.

Iterate towards an optimal solution Performing iterations to get an optimal solution as shown in Table below Since Cj-Ej is either zero or negative under all columns, the optimal basic feasible solution has been obtained.

First column represents coefficients of basic variables (current solution variables) in the objective (e) second column represent basic variables (current solution variables) and last column represents, right hand side of the constraints in standard form. after congesting all the inequalities into equalities, in any table, current values of the current solution variables (basic variables) is given by R. (Cj – Ej) represents the advantage of bringing any non basic variable to the current solution i.e. In the Table 2, values of Cj – Ej are 12, 15 and 14 for X.

If any of the values of Cj – Ej is ve then it means that most positive values represents the variable which is brought into current solution would increase the objective function to maximum extent. it would become basic and would enter the solution.

Initial Table becomes Since Cj-Ej is ve under some columns, solution given by Table 1 is not optimal.

It can be seen that out of -2 2M and -3 3M, -3 3M is most ve as M is a very large ve number. it can be seen that while solving problem with artificial variables, Cj-Ej row shows that optimal solution is reached whereas we still have artificial variable in the current solution having some vp value. Let us introduce artificial variables A, the above constraints can be written as.

Key element in Table 5 comes out to be 2 and it is made unity and all other elements in the key coloumn are made zero with the help of row operations and finally we get Table 6.

First key element is made unity by dividing that row by 2.

A system of linear inequalities defines a polytope as a feasible region.

The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimal solution.).

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