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## Alternate Optimal Solutions

Sometimes, when an optimization model is formulated, the model produces many alternative optimal solutions, meaning that for the same value of the objective function the model produces multiple values of the non-basic or decision variables.

Alternative optimal solutions mainly occur because some part of the polyhedron is parallel to the objective function. In such cases, all points along the segment of the part that is parallel to the obj function will be affine transformations and give the same value of the obj function.

In a practical situation, the implications of this would be that when you are trying to solve a problem, for example, you are trying to calculate the maximum profit given the effort to make 10 different products and the total labor constraint available on the floor. Assuming the problem has 10 decision variables and two constraints. Due to the degeneracy explained above, it can produce an optimal solution of maximum profit of USD 10,000 for multiple combinations of the product mix to be manufactured in the factory.

In such cases, it is very difficult to figure out which production mix to choose as the optimal criterion, since there are actually several values. The edge-parallel portion of the polyhedron or hyperplane connecting two planes of an n-dimensional polyhedron can be slightly altered by slightly adjusting the constraints.

The constraints of the linear programming model form the boundaries of the polyhedron or the hyperplane of the polyhedron. But just changing the constraint from, say, 4 * X + 5 * Y < 5 to 4 * X + 5 * Y < 5.1 would alter the feasible region a bit, but stop producing alternative optimal solutions.

In the same context we can also discuss what constitutes a convex feasible set and why linear programming problems require the constraint set to be convex. The optimal solution for a linear programming formulation is found by traversing the set of constraints from vertex to vertex. So why does an optimal solution not fall somewhere on an edge connecting two vertices, but only at the vertex?. This is because the feasible set can be visualized as the limit imposed by the constraints. Constraints in a linear programming model would result in a polyhedron/polytope. When this is convex it means that any point connecting the two vertices is not in the interior and therefore the extremal solution of the objective function will necessarily lie at the vertex.

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