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 $◻ \leq h \leq ◻$ .
(Simplify your answers.)

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Answer

The range of $h$ where the shadow price of labor does not change is $34 \leq h \leq 86$ .

Steps

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 $34 \leq h \leq 86$ .