(pen)o (g)Objective and Constraints(pen)
(pen)-> (g)Define variables(pen) and state their units.
(pen)-> (g)Constraints(pen) are inequalities the variables must satisfy.
(pen)-> The (g)feasible region(pen) is the overlap of all constraints.
(pen)-> The (g)objective(pen) is the quantity to maximize or minimize.
(pen)-> Example: maximize P = 3x + 2y.
(pen)-> Constraints: x >= 0, y >= 0, x + y <= 8, x <= 5.
(pen)-> Each (g)vertex(pen) is an intersection of two boundary lines.
(pen)-> A vertex must (g)satisfy every constraint(pen), not just two lines.
(pen)-> (g)Check(pen) candidate vertices in all constraints before use.
(pen)o (pu)Bounded Case(pen)
(pen)-> Example: maximize P = 4x + 5y.
(pen)-> Constraints: x >= 0, y >= 0, x + 2y <= 10, 3x + y <= 15.
(pen)-> (pu)Boundaries meet(pen) at (0, 0), (5, 0), (4, 3), and (0, 5).
(pen)-> (pu)Check(pen) (4, 3): 4 + 6 = 10 and 12 + 3 = 15.
(pen)-> (0, 0): P = 0.
(pen)-> (5, 0): P = 20.
(pen)-> (4, 3): P = 16 + 15 = 31.
(pen)-> (0, 5): P = 25.
(pen)-> The (pu)maximum(pen) is 31 at (4, 3).
(pen)o (br)Vertex Test(pen)
(pen)-> On a nonempty closed bounded polygon, a (br)linear objective(pen)
(pen)-> attains both (br)extremes(pen) at vertices; otherwise check existence.
(pen)-> (br)Evaluate every vertex(pen); do not stop at the first maximum.
(pen)-> Example: minimize C = 2x + 7y on the same region.
(pen)-> (0, 0): C = 0.
(pen)-> (5, 0): C = 10.
(pen)-> (4, 3): C = 8 + 21 = 29.
(pen)-> (0, 5): C = 35; the (br)minimum(pen) is 0 at (0, 0).
(pen)o (y)Unbounded Case(pen)
(pen)-> An (y)unbounded region(pen) may lack a maximum or minimum.
(pen)-> Example: minimize C = 2x + 3y.
(pen)-> Constraints: x >= 0, y >= 0, x + y >= 4.
(pen)-> The (y)vertices(pen) are (0, 4) and (4, 0).
(pen)-> C(0, 4) = 12 and C(4, 0) = 8.
(pen)-> At x = 0, C = 3y (y)grows without bound(pen), so no maximum exists.
(pen)-> Since x + y >= 4, C = 2(x + y) + y >= 8.
(pen)-> Thus the (y)minimum(pen) is 8 at (4, 0).
(pen)-> One extra test point cannot (y)prove(pen) that no maximum exists.
(pen)o (hp)Application(pen)
(pen)-> Each table needs 4 (hp)board feet(pen); each chair needs 2.
(pen)-> Only 24 board feet are (hp)available(pen).
(pen)-> At least 2 tables and at least 3 chairs are (hp)required(pen).
(pen)-> (hp)Profit(pen): $30 per table and $20 per chair.
(pen)-> (hp)Maximize(pen) P = 30t + 20c.
(pen)-> Constraints: 4t + 2c <= 24, t >= 2, c >= 3; t, c (hp)integers(pen).
(pen)-> Real-(hp)relaxation vertices(pen): (2, 3), (2, 8), and (4.5, 3).
(pen)-> (hp)Check(pen) (2, 3): 8 + 6 = 14 <= 24.
(pen)-> P values: 120, 220, and 195.
(pen)o (hp)Check and Interpret(pen)
(pen)-> The (hp)relaxed maximum(pen) is 220 at integer point (2, 8).
(pen)-> (hp)Integer points(pen) form a feasible subset of the relaxation.
(pen)-> So no (hp)feasible integer point(pen) can exceed 220.
(pen)-> (hp)Rounding(pen) a fractional optimal vertex is not a proof.
(pen)-> If needed, (hp)enumerate all feasible integer points(pen).
(pen)-> Report the number of tables, chairs, and (hp)profit(pen) with units.
(pen)-> State whether the region is (hp)bounded(pen) and the extreme claimed.
(pen)-> (hp)Confirm every constraint(pen) at the chosen point.