IGNOU AOR 1 2026 - Use dual simplex method to solve the following LPP:Minimize:Subj

Question

Use dual simplex method to solve the following LPP:
Minimize:
$$z = 3x_1 + x_2$
Subject to the constraints:
$$x_1 + x_2 \geq 1$
$$2x_1 + 3x_2 \geq 2$
$$x_1, x_2 \geq 0.$

09 Jan 2026

Answer :

Word Count : 804

We are asked to solve the LPP using the dual simplex method. Let’s solve it step by step manually. We have: Minimize: [ z = 3x_1 + x_2 ] Subject to: [ x_1 + x_2 \ge 1 ] [ 2x_1 + 3x_2 \ge 2 ] [ x_1, x_2 \ge 0 ] --- Step 1: Convert constraints to standard form for dual simplex For the dual simplex method, all constraints should be of the form (\le) for a minimization problem, but it’s more common to convert (\ge) constraints into (\le) by introducing surplus variables: [ x_1 + x_2 - s_1 = 1, \quad s_1 \ge 0 ] [ 2x_1 + 3x_2 - s_2 = 2, \quad s_2 \ge 0 ] Now the system becomes: [ x_1 + x_2 - s_1 = 1 ] [ 2x_1 + 3x_2 - s_2 = 2 ] Since we have surplus variables, the initial basic feasible solution (BFS) with (s_1, s_2) as basic variables is infeasible because (s_1 = -1, s_2 = -2). This is suitable for the dual simplex method because dual simplex starts with dual feasible but primal infeasible solution. --- Step 2: Set up initial tableau Variables: (x_1, x_2, s_1, s_2) Basic variables: (s_1, s_2) Initial tableau: | BV | (x_1) | (x_2) | (s_1) | (s_2) | RHS | | ----- | ----- | ----- | ----- | ----- | --- | | (s_1) |

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