Consider the problem of manufacturing widgets to maximize revenue,
subject to labor and material availability:
max 4x1 + x2
x1 + x2 ≤ 2 (labor)
3x1 + x2 ≤ 3 (materials)
x1, x2 ≥ 0
where x1, x2 represent the production levels of deluxe and regular widgets,
respectively.
(a) Solve the problem by inspecting the graph.
(b) Compute B−1 and the shadow prices of labor and materials. Note that
xB is (x3, x1), not (x1, x3), so that B =
1 1
0 3
.
(c) Suppose an additional six cases of materials are made available. Use the
shadow price of materials to derive an upper bound on the new objective
function. What is the actual value of the new objective function?