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