Question

1) Find the vertices of the feasible region. 2) What is the maximum and the minimum value of the function Q = 70x + 82y on the feasible region?

Answer 1

To find the vertices of the feasible region, graph the inequalities and identify the points where the lines intersect. To find the maximum and minimum value of the function, substitute the x and y values of the vertices into the function and compare the values.

Step-by-step explanation:

To find the vertices of the feasible region, we need to graph the inequalities and identify the points where the lines intersect. The feasible region is the area that satisfies all the given inequalities. In this case, the inequalities are 0 ≤ x ≤ 20 and 10 ≤ y ≤ 20. By graphing these inequalities, we can identify the four vertices as (0,10), (0,20), (20,10), and (20,20).

To find the maximum and minimum value of the function Q = 70x + 82y on the feasible region, substitute the x and y values of the vertices into the function and calculate the corresponding values of Q. Compare these values to determine the maximum and minimum.

Answer 2

If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore, if the objective function P is optimized at two adjacent vertices of S, then it is optimized at every point on the line segment joining these two vertices, in which case there are infinitely many solutions to the problem. Suppose we are given a linear programming problem with a feasible set S and an objective function P = ax+by. Then, If S is bounded then P has both a maximum and minimum value on S If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided that the constraints defining S include the inequalities x≥ 0 and y≥ 0. If S is the empty set, then the linear programming problem has no solution; that is, P has neither a maximum nor a minimum value.