July 12, 2024

Network System

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Alternate Best Solutions

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Alternate Best Solutions

Often when an optimization product is formulated the product yields a good deal of alternate exceptional remedies indicating that for the exact same worth of the goal function the model yields many value of the non fundamental or determination variables.

Alternate ideal answers take place primarily due to some part of the polyhedron remaining parallel to the goal purpose. In these kinds of cases all details alongside the section of the portion that is parallel to the obj operate will be affine transforms and would produce the very same benefit of the obj functionality.

In a useful circumstance the implications of this would be that when one particular is seeking to remedy a issue of say attempting to calculate the optimum earnings supplied the exertion to manufacture 10 distinct items and the overall constraint on accessible labour in the plant. Supposing the challenge has 10 final decision variables and two constraints. Due to degeneracy defined previously mentioned it might generate an best option of greatest financial gain of USD 10000 for a number of combinations of the product or service blend expected to be created in the manufacturing facility.

In this kind of instances it is very tricky to determine out which production blend to decide on as the optimal criterion as there are in fact various values. The parallel portion of the possibly the edge of the polyhedron or the hyperplane that connects two planes of an n dimensional polyhedron can be disturbed a bit by tweaking the constraints a very little bit.

The constraints in the linear programming model kind the boundaries of the polyhedron or the hyper airplane of the polyhedron. But just modifying the constraint say from 4*X + 5 * Y < 5 to 4*X + 5 * Y < 5.1 would result in altering the feasible region just little bit, but would cease to produce alternate optimal solutions.

In the same context we can also discuss what forms a feasible convex set and why linear programming problems require the set of constraints to be a convex. The optimal solution to a linear programming formulation is found out by traversing the set of constraints from vertex to vertex. So why does an optimal solution not fall somewhere on an edge that connects two vertices, but only on the vertex?. This is because the feasible set can be visualized as the boundary enforced by constraints. The constraints in a linear programming model would result in a polyhedron /polytope. When this is convex it means that any point connecting the two vertexes does not lie inside and so the extreme solution of the objective function will be necessarily found on the vertex.

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