Verify whether the following points are optimal solutions to the LP?

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In summary: HJpY3Rpb24gSW5zY3JpcHRpb24gLSBQb2ludHMgKDQsNCkgYW5kIChDKDApLCBhbmQgKDIsMCkpCg1NbmltaXNlIDN4MSs2eDIgKyA2eDIgKyAxMiAqIDEyCnMuIFN0byBNU0QgMjEvNjJ4MS0zNXg2PTQKIn summary, the question asks to minimize the expression 3x1+6x2 with the constraints 6x1-3x2=12 and x1, x
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ashina14
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Homework Statement



Points (4,4) and (2,0)

Minimise 3x1+6x2
s.t. 6x1-3x2=12
x1,x2>=0



Homework Equations





The Attempt at a Solution



I tried solving this the way LP questions are solved in general, graphically.
So I drew a graph plotting the objective and the constraint but turns out there aren't enough constrains to have a fixed feasible region?! Either I move the objective along points on the constraint line only. Or there is something completely different required to do here?
 
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  • #2
ashina14 said:

Homework Statement



Points (4,4) and (2,0)

Minimise 3x1+6x2
s.t. 6x1-3x2=12
x1,x2>=0



Homework Equations





The Attempt at a Solution



I tried solving this the way LP questions are solved in general, graphically.
So I drew a graph plotting the objective and the constraint but turns out there aren't enough constrains to have a fixed feasible region?! Either I move the objective along points on the constraint line only. Or there is something completely different required to do here?

This is wrong. You DO have a feasible region (I do not know what you mean by a "fixed" feasible region). The feasible region in this case happens to be unbounded (that is, contains points (x1,x2) where both x1 and x2 go to +∞) but that does not matter. In this case there is a unique minimizing point (so there are not two optimal points, just one).

RGV
 

FAQ: Verify whether the following points are optimal solutions to the LP?

1. What is an optimal solution in LP?

An optimal solution in LP (Linear Programming) is the combination of decision variables that maximizes or minimizes the objective function while satisfying all the constraints. It is the best possible solution to the problem at hand, and there can be one or multiple optimal solutions depending on the LP problem.

2. How do you verify if a point is an optimal solution in LP?

To verify whether a point is an optimal solution in LP, we need to check if it satisfies all the constraints of the problem and if it gives the best possible value for the objective function. This can be done by substituting the values of the decision variables into the objective function and comparing it with the values of the objective function at other feasible points. If the point in question gives the highest or lowest value, depending on the objective function, and satisfies all the constraints, then it is an optimal solution.

3. Can a point be an optimal solution if it violates some constraints?

No, a point cannot be an optimal solution if it violates any constraints. The whole purpose of LP is to find a feasible solution that satisfies all the constraints. If a point violates any constraint, then it is not a feasible solution and cannot be considered as an optimal solution.

4. Can there be more than one optimal solution in LP?

Yes, there can be more than one optimal solution in LP. This happens when the objective function has a linear relationship with the decision variables, and there are multiple combinations of decision variables that give the same optimal value for the objective function. In such cases, any of the optimal solutions can be chosen as the final solution.

5. How does one determine the optimality of a solution in LP?

The optimality of a solution in LP can be determined by calculating the sensitivity of the objective function to the changes in the decision variables. If a small change in the decision variables leads to a significant change in the objective function value, then the solution is not optimal. However, if the objective function is not sensitive to the changes in the decision variables, then the solution can be considered optimal.

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