Simplex Table

        Computational procedures in the simplex method require the construction of the simplex table which can be done in different ways. All of which however, led to the same optimum solution. The initial simplex table is formed by writing out the coefficients and constraints of linear programming problem in a systematic tabular form. The rules used for the construction of the initial simplex table are same in both the maximization and minimization problems.

        Consider the general linear programming problem as stated below:

                                     Maximize Z = C₁X + C₂X₂ +......+C_nX_n

                                    Subject to the constraints

                                    A₁₁X₁ + a₁₂X₂ +......+ a1_n x_n< b₁

                                                A₂₁X₁ + a₂₂x2 +.....+a_2nx_n> b₂

                                                Am₁X₁+ a_m2X₂ +...... + a_mnX_n< b _m

And                                                                                   X₁,X₂,...., X_n >0

Introducing slack variables to convert inequalities into equations the above problem reduces to the form:

Maximize Z = C₁X₁ + C₂ X₂ + ......+ C_n X_n +0. S₁ +0. S₂ + ...+0. S_m

Subject to the constraints

                                    A₁₁X₁ + a₁₂ X₂ +.....+ a1_nX_n + S₁ = b₁

                                    A₂₁ X₁+ a₂₂X₂+....+ a_2nX_n +S₂ = b₂

                                                A_m1 X₁ + a_m2X₂ +....+ a_mnX_n + S_m = b_m

And                                  x1,x2......, x_n; s1, s2,...., s_m > 0

The first row, called the objective row of the simplex table indicates the values of C_j (j subscripts refer to the column number) which are the coefficients of the (m+n) variables in the objective function. These coefficients are obtained directly from the objective function and the values of C_j would remain the same in the succeeding tables. The second row of the table provides the major column headings for the table and these column headings remain unchanged

            The first column C_b called the objective column are the coefficients of the current basic variables in the objective function. The second column B (known as product mix column) point out the basic variables in the basis, and in the initial simplex table these basic variables are the slack variables. The third column (quantity column) b=X_B indicates the resources or the solution values of the basic variables.

The identity matrix in the simplex table represent the coefficients of the slack variables that have been added to the original inequalities to make them equations. Each simplex table contains an identity matrix under the basic variables. The matrix under non-basic variables in the simplex table is called coefficient matrix. The number a_ij in the coefficient matrix will either be negative zero or positive.

            The row labeled Z_ij contains the sum of the products of the numbers in the C_B column times the corresponding under each column variable in the main body of the table. The value of Z_j under each column variable represents the amount of profit loss (opportunity cost) for each unit variable that is brought into solution at the current iteration.

            The final row, C_j - Z_j row called the index row or net evaluation row is determined by subtracting the Z_j value from the corresponding C_j value. Each number in the index row represents the net marginal improvement on the objective function if one unit of each variable x_j is brought into the solution as the current iteration.

            Finally the value of the objective function for the current solution (given in the right bottom of the table) is determined by the sum of the products of the number X_B column and the corresponding numbers in the C_B column.

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