Problem
A company manufactures paring knives and pocket knives. Each paring knife requires 3 labor-hours, 7 units of steel, and 4 units of wood. Each pocket knife requires 6 labor-hours, 5 units of steel, and 3 units of wood. The profit on each paring knife is $\$ 3$, and the profit on each pocket knife is $\$ 5$. Each day the company has available 102 labor-hours, 157 units of steel, and 114 units of wood. Suppose that the number of labor-hours that are available each day is increased by $h$. For what values of $h$ will a change of $h$ labor-hours not change the shadow price of labor? The shadow price of labor will not change for $\square \leq h \leq \square$. (Simplify your answers.)
Solution
Step 1 ：Set up a linear programming model to maximize profit with the given constraints.
Step 2 ：Find the shadow price of labor, which is the dual value of the labor constraint in the linear programming problem.
Step 3 ：Solve the dual problem and observe how the dual value for the labor constraint changes as we vary \( h \).
Step 4 ：Determine the range of values for \( h \) where the shadow price of labor remains constant.
Step 5 ：The range of \( h \) where the shadow price of labor does not change is \(\boxed{34 \leq h \leq 86}\).
