Consider the following LPP, which is being solved by the dual simplex method. Maximize $z = -x_1 + x_2$ subject to $x_1 - 4x_2 \ge 5$ $x_1 - 3x_2 \le 1$ $2x_1 - 5x_2 \ge 1$ $x_1, x_2 \ge 0$ Note: Answer based on the rules for entering and leaving variables of the dual simplex method. Which of the following is true for the given problem? Select all that apply. The entering variable for the first iteration is $x_1$. The optimum solution is obtained at $x_1 = 5, x_2 = 0$. There is no feasible solution for the given LPP. The optimal value of the objective function is -5.
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Step 1: First, we convert the inequalities into equalities by introducing slack variables $s_1, s_2, s_3$: $x_1 - 4x_2 - s_1 = 5$ $x_1 - 3x_2 + s_2 = 1$ $2x_1 - 5x_2 - s_3 = 1$ $x_1, x_2, s_1, s_2, s_3 \ge 0$ Show more…
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Madhur L.
Optimality condition: The entering variable in a maximization (minimization) problem is the nonbasic variable having the most negative (most positive) coefficient in the z-row. The optimum is reached at the iteration where all the z-row coefficients of the nonbasic variables are nonnegative (nonpositive). Feasibility condition: For both the maximization and the minimization problems, the leaving variable is the basic variable associated with the smallest nonnegative ratio between the elements in solution column and the pivot column. Consider the following set of constraints, solve the problem using the simplex method Maximize z = 3x1 - x2 + 3x3 + 4x4 subject to x1 + 2x2 + 2x3 + 4x4 <= 40 2x1 - x2 + x3 + 2x4 <= 8 4x1 - 2x2 + x3 - x4 <= 10 x1, x2, x3, x4 >= 0
The following tableau represents a specific simplex iteration. All variables are nonnegative. The tableau is not optimal for either a maximization or a minimization problem. Thus, when a nonbasic variable enters the solution, it can either increase or decrease z or leave it unchanged, depending on the parameters of the nonbasic variable. z x1 x2 x3 x4 x5 x6 x7 x8 RHS Basic 1 0 -5 0 4 -1 -10 0 0 620 z 0 0 3 0 -2 -3 -1 5 1 12 0 0 1 1 3 1 0 3 0 6 0 1 -1 0 0 6 -4 0 0 0 a) Categorize the variables as basic and nonbasic, and provide the current values of all the variables. b) Suppose the problem is of the maximization type; identify the nonbasic variables that have the potential to improve the value of z. If each such variable enters the basic solution, determine the associated leaving variable, if any, and the associated change in z. Do not use the elementary row operations. c) Repeat part (b) assuming that the problem of minimization type. d) Which nonbasic variable(s) will not cause a change in the value of z when selected to enter the solution?
Andreas P.
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