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## Alternate Optimal Solutions

Sometimes, when an optimization model is formulated, the model produces many alternative optimal solutions, which means that for the same value of the objective function, the model produces a multiple value of the nonfundamental or decision variables.

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

In a practical situation the implications of this would be that when trying to solve a problem, say trying to calculate the maximum profit given the effort of making 10 different products and the total stress on labor work available in the factory. Suppose the problem has 10 decision variables and two constraints. Due to the degeneracy explained above, this can give an optimal solution of maximum profit of $10,000 for several combinations of the product line to be produced in the factory.

In such cases, it is very difficult to determine which production mix to choose as the optimal criterion because there are actually several values. The parallel part of the polyhedron or hyperplane edge that connects two planes of an n-dimensional polyhedron can be slightly disturbed by adjusting the constraints a bit.

The constraints of the linear programming model form the boundaries of the polyhedron or the polyhedron’s hyperplane. But simply changing the constraint, say, from 4*X + 5 * Y < 5 to 4*X + 5 * Y < 5.1, would cause the feasible region to change minimally, but would cease to produce alternative optimal solutions.

In the same context, we can also discuss what forms a feasible convex set and why linear programming problems require the constraint set to be a convex one. The optimal solution of a linear programming formulation is found by traversing the set of constraints from vertex to vertex. So why doesn’t an optimal solution fall somewhere on an edge that connects two vertices, but only on the vertex?. Indeed, the feasible set can be visualized as the boundary imposed by constraints. Constraints in a linear programming model would result in a polyhedron/polytope. When the latter is convex, this means that any point connecting the two vertices is not inside and that the extreme solution of the objective function will therefore necessarily be on the vertex.

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