Question

True or False?

1. If an LP has multiple optimal solutions, then all solutions have the same objective function value (such as total profit or cost).

2. As long as all prices (objective coefficients) change within their respective ranges, the optimal solution of a linear program does not change.

3. When a dual price changes within its range, the optimal solution does not change.

4. The solution of a linear program always consists of whole numbers (integers). That is why the linear program is so popular in practice.

5. If a constraint is not tight then its dual price has to be zero. This conclusion is true for both >= and <= constraints.

6. If the reduced cost of a product (decision variable) is positive, then the objective function will decrease by the amount of the reduced cost if an additional such product is produced.

Answer 1

answer..

1.

True.

This is true because if the inverse was true and there were two optimal values, then only one of them can be optimal and therefore only one can be an optimal solution. Suppose for a maximization problem, you have two solutions one with objective function = 5 units and one with objective function 10 units, then automatically the optimal solution referring to objective function equal to 10 units becomes the optimal solution and one with 5 units cannot be optimal by definition.

2.

True – That is the essential meaning of the change of objective coefficient in respective ranges. While the objective function value at optimal can still change.

5.

True – The dual price is the improvement in objective function (reduction in minimization problem and increase in maximization problem by relaxing one unit of a constraint RHS.