![]() Hence proceed to improve this solution in step 4. (iii) If corresponding to most negative Δ j, all elements of the column X j are negative or zero (≤ 0), then the solution under test will be unboundedĪpplying this rule for testing the optimality of starting basic feasible solution, it is observed that Δ 1, and Δ 2 both are negative. (ii) If at least one Δ j is negative, the solution under test is not optimal, then proceed to improve the solution in step 4. (i) If all Δ j ≥ 0, the solution under test will be optimal. This is done by computing the “net evaluation” D j for variable x j by the formula Now, proceed to test basic feasible for optimality by the rules given below. So 5, = 4, s 2 = 2, here The complete starting feasible solution can be immediately read from table 2 as s 1 = 4, s 2, x, = 0, x 2 = 0 and the value of the objective function is zero. So x 1 = x 2 = 0 here, column x B gives the values of basic variables in the first column. It should be remembered that values of non-basic variables are always zero at each iteration. The values z j represents the amount by which the value of objective function Z would be decreased or increased if one unit of given variable is added to the new solution. ![]() ![]() Number a ij represent the rate at which resource (i- 1, 2- m) is consumed by each unit of an activity j (j = 1,2 … n). Column x B gives the current values of the corresponding variables in the basic. Column C B gives the coefficients of the current basic variables in the objective function. The second row gives major column headings for the simple table. These values represent cost or profit per unit of objective function of each of the variables. The first row in table indicates the coefficient c j of variables in objective function, which remain same in successive tables. Write down the coefficients of all the variables in given LPP in the tabular form, as shown in table below to get an initial basic Feasible solution. In this example, the inequality constraints being ‘≤’ only slack variables s 1 and s 2 are needed. The coefficients of slack or surplus variables are zero in the objective function. (iv) Next convert the inequality constraints to equation by introducing the non-negative slack or surplus variable. In this example, all the b i (height side constants) are already positive. If not, it can be changed into positive value on multiplying both side of the constraints by-1. (iii) Ensure all b i values are positive. (ii) If objective function is of minimisation type then convert it into one of maximisation by following relationship (i) Formulate the mathematical model of given LPP. The steps in simplex algorithm are as follows: ![]()
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