Math Problem Statement

Solve

Maximize Z = 2x1 + x2 under the constraints -x1 + x2 ≤ 1, x1 - 2x2 ≤ 2, x1, x2 ≥ 0.

Solution

To solve this linear programming problem, we aim to maximize Z=2x1+x2Z = 2x_1 + x_2 under the given constraints.

Problem Statement

Objective function: Maximize Z=2x1+x2\text{Maximize } Z = 2x_1 + x_2

Subject to the constraints: x1+x21-x_1 + x_2 \leq 1 x12x22x_1 - 2x_2 \leq 2 x1,x20x_1, x_2 \geq 0

Step-by-Step Solution

  1. Convert inequalities into boundary lines:

    • For x1+x2=1-x_1 + x_2 = 1: x2=x1+1x_2 = x_1 + 1
    • For x12x2=2x_1 - 2x_2 = 2: x2=x122=0.5x11x_2 = \frac{x_1 - 2}{2} = 0.5x_1 - 1
  2. Identify the feasible region:

    • The inequalities define a feasible region in the first quadrant (x1,x20x_1, x_2 \geq 0) where the lines x2=x1+1x_2 = x_1 + 1 and x2=0.5x11x_2 = 0.5x_1 - 1 intersect or bound the region.
  3. Find the points of intersection:

    • Set x1+1=0.5x11x_1 + 1 = 0.5x_1 - 1 to find the intersection of the two lines: x10.5x1=2x_1 - 0.5x_1 = -2 0.5x1=20.5x_1 = 2 x1=4x_1 = 4
    • Substitute x1=4x_1 = 4 into x2=x1+1x_2 = x_1 + 1: x2=4+1=5x_2 = 4 + 1 = 5
    • Thus, the point of intersection is (4,5)(4, 5).
  4. Check the vertices of the feasible region:

    • The vertices of the feasible region (where the constraints are satisfied) include:
      • Intersection with x1=0x_1 = 0:
        • x2=0+1=1x_2 = 0 + 1 = 1 from x2=x1+1x_2 = x_1 + 1, so we have (0,1)(0, 1).
      • Intersection with x2=0x_2 = 0:
        • x1=2x_1 = 2 from x12x2=2x_1 - 2x_2 = 2, so we have (2,0)(2, 0).
      • The point (4,5)(4, 5) where the two lines intersect.
  5. Calculate Z=2x1+x2Z = 2x_1 + x_2 at each vertex:

    • At (0,1)(0, 1): Z=2(0)+1=1Z = 2(0) + 1 = 1
    • At (2,0)(2, 0): Z=2(2)+0=4Z = 2(2) + 0 = 4
    • At (4,5)(4, 5): Z=2(4)+5=8+5=13Z = 2(4) + 5 = 8 + 5 = 13
  6. Conclusion: The maximum value of Z=2x1+x2Z = 2x_1 + x_2 is achieved at (4,5)(4, 5) with Z=13Z = 13.

Final Answer

Maximum Z=13 at (x1,x2)=(4,5)\text{Maximum } Z = 13 \text{ at } (x_1, x_2) = (4, 5)

Would you like more details on any specific part, or do you have further questions?


Additional Questions:

  1. What would change if we were to minimize Z=2x1+x2Z = 2x_1 + x_2 instead?
  2. How would the solution be affected if the inequality x1+x21-x_1 + x_2 \leq 1 was replaced by x1+x21-x_1 + x_2 \geq 1?
  3. Can you visualize the feasible region on a graph to confirm the vertices?
  4. How does the choice of objective function coefficients (e.g., 2x1+x22x_1 + x_2) influence the maximum point?
  5. What happens if we add an additional constraint, like x1+x26x_1 + x_2 \leq 6?

Tip: Graphical visualization of constraints can greatly help in confirming the feasible region and solution in linear programming.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization
Inequalities

Formulas

Objective function: Z = 2x1 + x2

Theorems

Fundamental Theorem of Linear Programming

Suitable Grade Level

Grade 11-12