**Alternative Optimal Solutions****.**While discussing the corner points method it was mentioned that an optimal solution will always occur at a corner point on the area of feasible solution. There is a possibility of more than one optimal solutions in linear programming problems. The following figure illustrates a case where the objective function has the same slope as one of the constraint. If the objective function is improved by moving out, away from the origin, the last points touched before the function moves outside the area of feasible solution are all points on line AB.

In this situation, there would exist an infinite number of points each resulting in the same maximum value of Z. For such situations, we say that there are alternative optimal solutions to the problem.

For alternative optimal solutions to exist, two conditions need to be satisfied.

- The objective function must be parallel to the constraint which forms an edge or boundary on the area of feasible solution.
- The constraint must form a boundary on the area of feasible solution in the direction of the optimal movement of the objective function, i.e. the constraints must be a binding constraint preventing further improvement in the value of the objective function. In the above figure, the second condition would be violated if the problem was one of minimisation i.e. if we wanted to shift the objective function in the other direction.

When using the comer point method, alternative optimal solutions are indicated if a tie occurs for the optimal value of the objective function. The alternative optimal solution occurs at the ‘tying’ comer points, as well as along the entire line connecting the two points.

**Sol**. The above constraints are plotted as shown in the figure below.

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