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After graphing the equations for the linear programming problem, the graph will have some intersecting lines forming some polygon. This polygon (triangle, rectangle, parallelogram, quadrilateral, etc) is the feasible region.
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What is the feasible region in linear programming?
Linear programming is just graphing a bunch of linear inequalities. Remember that when you graph inequalities, you need to shade the "good" region - pick a point that is not on the line, put it in the inequality, and the it the point makes the inequality true (like 0
What is the minimum value of 6x plus 5y in the feasible region excluding point (0 0)?
The answer depends on the feasible region and there is no information on which to determine that.
What is the maximum value of 3x 4y in the feasible region?
240
What is the maximum value of 4x 3y in the feasible region?
26
What is the maximum value of 5x plus 2y in the feasible region?
42
Is it possible for a linear programming problem to have no solution?
Yes. There need not be a feasible region.
Distinguish between integer programming problem and linear programming problem?
Integer programming is a subset of linear programming where the feasible region is reduced to only the integer values that lie within it.
Can a linear programming problem have multiple solutions?
Yes. If the feasible region has a [constraint] line that is parallel to the objective function.
What is optimal feasible solution?
It is usually the answer in linear programming. The objective of linear programming is to find the optimum solution (maximum or minimum) of an objective function under a number of linear constraints. The constraints should generate a feasible region: a region in which all the constraints are satisfied. The optimal feasible solution is a solution that lies in this region and also optimises the obective function.
What is the feasible region in linear programming?
Linear programming is just graphing a bunch of linear inequalities. Remember that when you graph inequalities, you need to shade the "good" region - pick a point that is not on the line, put it in the inequality, and the it the point makes the inequality true (like 0
What is optimal solution?
It is usually the answer in linear programming. The objective of linear programming is to find the optimum solution (maximum or minimum) of an objective function under a number of linear constraints. The constraints should generate a feasible region: a region in which all the constraints are satisfied. The optimal feasible solution is a solution that lies in this region and also optimises the obective function.
How do you get a feasibility region linear programming?
A feasible region is, in a constrained optimization problem, the set of solutions satisfying all equalities and/or inequalities. On the other hand a linear programming is a constrained optimization problem in which both the objective function and the constraints are linear, therefore a feasible region on a linear programming problem is the set of solutions of the a linear problem. Many algorithms had been designed to successfully attain feasibility at the same time as resolving the problem, e.g. reaching its minimum. Perhaps one of the most famous and extensively utilized is the Simplex Method who travels from one extremal point to another, which happens to be the possible extrema given the convex nature of the problem, by maintaining a fixed number of components to zero, called basic variables. Then, the algorithm arrives to a global minimum generally in polinomial time even if its worst possible case has already been proved to be exponencial, see Klee-Minty's cube.
What is the definition of the solution of linear inequalities?
Each linear equation is a line that divides the coordinate plane into three regions: one "above" the line, one "below" and the line itself. For a linear inequality, the corresponding equality divides the plane into two, with the line itself belonging to one or the other region depending on the nature of the inequality. A system of linear inequalities may define a polygonal region (a simplex) that satisfies ALL the inequalities. This area, if it exists, is called the feasible region and comprises all possible solutions of the linear inequalities. In linear programming, there will be an objective function which will restrict the feasible region to a vertex or an edge of simplex. There may also be a further constraint - integer programming - where the solution must comprise integers. In this case, the feasible region will comprise all the integer grid-ponits with the simplex.
Why are integer programming problems more difficult to solve than linear programming problems?
In both cases the constraints are used to produce an n-dimensional simplex which represents the "feasible region". In the case of linear programming this is the feasible region. But that is not the case for integer programming since only those points within the region for which the variables are integer are feasible.The objective function is then used to find the maximum or minimum - as required. In the case of a linear programming problem, the solution must lie on one of the vertices (or along one line in 2-d, plane in 3-d etc) of the simplex and so is easy to find. In the case of integer programming, the optimal solution so found may contain one or more variables that are not integer and so it is necessary to examine all the points in the immediate neighbourhood and evaluate the objective function at each of these points. This last requirement makes integer programming solutions more difficult to find.
Will points in a feasible region be a solution to the real-world problem it represents?
Yes they will. That is how the feasible region is defined.
Is (-10) a linear inequality?
II. SIMPLEX ALGORITHM A. Primal Simplex Algorithm If the unconstrained solution space is defined in n dimensions (each dimension assumed to be infinite), each inequality constraint in the linear programming formulation divides the solution space into two halves. The convex shape defined in n-dimensional space after m bisections represents the feasible area for the problem, and all points which lie inside this space are feasible solutions to the problem. Figure 1 shows the feasible region for a problem defined in two variables, n = 2, and three constraints, m = 3. Note that in linear programming, there is an implicit non-negativity constraints for the variables. The linearity of the objective function implies that the the optimal solution cannot lie within the interior of the feasible region and must lie at the intersection of at least n constraint boundaries. These intersections are known as corner- point feasible (CPF) solutions. In any linear programming problem with n decision variables, two CPF solutions are said to be adjacent if they share n − 1 common constraint boundaries. When interpreted geometrically, the Simplex algorithm moves from one corner-point feasible solution to a better corner-point-feasible solution along one of the constraint boundaries. There are only a finite number of CPF solutions, although this number is potentially exponential in n, however it is not necessary to visit all of them to determine the optimal solution to the problem. The convex nature of linear programming means that there are no local maxima present in the problem which are not also global maxima. Hence if at some CPF solution, no improvement is made by a move to another adjacent CPF then the algorithm terminates and we can be confident that the optimal solution has been found.
What is primal simplex method?
II. SIMPLEX ALGORITHM A. Primal Simplex Algorithm If the unconstrained solution space is defined in n dimensions (each dimension assumed to be infinite), each inequality constraint in the linear programming formulation divides the solution space into two halves. The convex shape defined in n-dimensional space after m bisections represents the feasible area for the problem, and all points which lie inside this space are feasible solutions to the problem. Figure 1 shows the feasible region for a problem defined in two variables, n = 2, and three constraints, m = 3. Note that in linear programming, there is an implicit non-negativity constraints for the variables. The linearity of the objective function implies that the the optimal solution cannot lie within the interior of the feasible region and must lie at the intersection of at least n constraint boundaries. These intersections are known as corner- point feasible (CPF) solutions. In any linear programming problem with n decision variables, two CPF solutions are said to be adjacent if they share n − 1 common constraint boundaries. When interpreted geometrically, the Simplex algorithm moves from one corner-point feasible solution to a better corner-point-feasible solution along one of the constraint boundaries. There are only a finite number of CPF solutions, although this number is potentially exponential in n, however it is not necessary to visit all of them to determine the optimal solution to the problem. The convex nature of linear programming means that there are no local maxima present in the problem which are not also global maxima. Hence if at some CPF solution, no improvement is made by a move to another adjacent CPF then the algorithm terminates and we can be confident that the optimal solution has been found.
