Maximize Z = 3×1 + 2×2

Problem 4. (25 Points)
Solve the following problem graphically (Please be neat). Draw the polytope on the x-y coordinate system (can be done either by hand or computer). Show all intersection of the polytope and identify the point (x,y coordinate) where the objective function is maximized and provide that value.

Maximize Z = 3×1 + 2×2
Subject to:
1×1 + 1×2 ≤ 10
8×1 + 1×2 ≤ 24
and
x1, x2 ≥ 0

Problem 5. (30 Points)
Work through the simplex method (in algebraic form) step by step to solve the following problem. Show all work and provide the solutions for each variable at every iteration of the simplex.

Maximize z = 4×1 + 3×2 + 4×3

Subject to:
2×1 + 2×2 + 1×3 ≤ 20
2×1 + 1×2 + 2×3 ≤ 14
1×1 + 1×2 + 3×3 ≤ 15
and
x1, x2, x3 ≥ 0