A feasible solution to an lp problem
a) Must satisfy all of the problems constraints simultaneously
b) Need not satisfy all of the constraints, only some of them
c) Must be a corner point of the feasible region
d) Must optimize the value of the objective function
The correct answer and explanation is :
Correct Answer: a) Must satisfy all of the problem’s constraints simultaneously
Explanation:
In Linear Programming (LP), a feasible solution refers to any solution that satisfies all the constraints of the problem, including both the equality/inequality constraints and the non-negativity constraints (if applicable).
Let’s break this down:
1. Feasible Region
An LP problem is defined by:
- An objective function (to be maximized or minimized),
- A set of constraints (usually linear inequalities or equalities).
The feasible region is the set of all points (combinations of variable values) that satisfy all these constraints simultaneously. A feasible solution is any point within this region.
2. Why Option (a) is Correct
For a solution to be feasible, it must:
- Not violate any constraint,
- Lie within the feasible region.
If a point fails to meet even one constraint, it lies outside the feasible region and is considered infeasible.
3. Why the Other Options are Incorrect
- (b): Saying it “need not satisfy all of the constraints” contradicts the definition of feasibility. Violating any constraint makes the solution infeasible.
- (c): While optimal solutions often occur at corner points (extreme points) of the feasible region (in linear problems), not all feasible solutions are corner points. Feasible solutions can lie anywhere in the feasible region, including along the edges or in the interior.
- (d): A feasible solution does not necessarily optimize the objective function. It’s simply a candidate. Among all feasible solutions, the one that gives the best value of the objective function is the optimal solution.
Summary
A feasible solution is any point that satisfies all constraints of the LP problem. It’s a basic requirement, not necessarily the best (optimal) solution. Finding a feasible solution is the first step before seeking optimality.
