Introduction to unbounded solution:
Unbounded solution of an objective function is a feasible set of points, which are unbounded in a particular direction. An unbounded solution may or may not have a maximum or a minimum value and if it has a maximum or a minimum then it occurs at the extreme points. Unbounded region have infinite set of solutions.
Unbounded solution set is feasible and may extend beyond positive or negative infinity. There can be unbounded solution for both maximizing as well as minimizing problems. Only the solution or set of points can be unbounded but the constraints, which are defined, are never unbounded. Simply put unbounded solutions are not in an enclosed area.
Modes of Unbounded Solution
FOR MAXIMIZING PROBLEM: When maximizing is to be done the solution set may have an infinitely large value, which makes the solution set unbounded at the positive infinity.
FOR MINIMIZING PROBLEM: When minimizing is to be done the solution set may have infinitely small values, which make the solution, set unbounded at the negative end.
Unbounded Solution-causes and Illustrations
Causes: One of the major causes of unbounded solution is the improper formulation of the problem. Unbounded solution set of a problem may occur if one of the constraints of the problem is inadvertently removed.
An unbounded solution can be converted to a bounded one by changing the objective function. Real world problems usually do not have unbounded solutions. Sometimes an unbounded region may not have an optimal solution. Addition of a constraint can also make an unbounded region into a bounded one. Inorder to find an optimal solution of an unbounded solution usually z line is drawn and no solution is considered to be optimal beyond the z line.
Illustration: Minimize C = 3x + 4y subject to the constraints
3x - 4y ≤ 12,
x + 2y ≥ 4
x ≥ 1, y ≥ 0.
The feasible region of this problem is unbounded with points at (1,1.5) and (4,0)
Although the feasible region is unbounded, we can minimize C = 3x + 4y at x=1,y=1.5 so that C=9
Unbounded solution of an objective function is a feasible set of points, which are unbounded in a particular direction. An unbounded solution may or may not have a maximum or a minimum value and if it has a maximum or a minimum then it occurs at the extreme points. Unbounded region have infinite set of solutions.
Unbounded solution set is feasible and may extend beyond positive or negative infinity. There can be unbounded solution for both maximizing as well as minimizing problems. Only the solution or set of points can be unbounded but the constraints, which are defined, are never unbounded. Simply put unbounded solutions are not in an enclosed area.
Modes of Unbounded Solution
FOR MAXIMIZING PROBLEM: When maximizing is to be done the solution set may have an infinitely large value, which makes the solution set unbounded at the positive infinity.
FOR MINIMIZING PROBLEM: When minimizing is to be done the solution set may have infinitely small values, which make the solution, set unbounded at the negative end.
Unbounded Solution-causes and Illustrations
Causes: One of the major causes of unbounded solution is the improper formulation of the problem. Unbounded solution set of a problem may occur if one of the constraints of the problem is inadvertently removed.
An unbounded solution can be converted to a bounded one by changing the objective function. Real world problems usually do not have unbounded solutions. Sometimes an unbounded region may not have an optimal solution. Addition of a constraint can also make an unbounded region into a bounded one. Inorder to find an optimal solution of an unbounded solution usually z line is drawn and no solution is considered to be optimal beyond the z line.
Illustration: Minimize C = 3x + 4y subject to the constraints
3x - 4y ≤ 12,
x + 2y ≥ 4
x ≥ 1, y ≥ 0.
The feasible region of this problem is unbounded with points at (1,1.5) and (4,0)
Although the feasible region is unbounded, we can minimize C = 3x + 4y at x=1,y=1.5 so that C=9
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