Linear Programming Math Example 1

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Example 1

medium
Maximize z=3x+2yz = 3x + 2y subject to x+y≤4x + y \leq 4, x≥0x \geq 0, y≥0y \geq 0.

Solution

  1. 1
    Step 1: Identify the feasible region: the triangle with vertices (0,0)(0,0), (4,0)(4,0), (0,4)(0,4).
  2. 2
    Step 2: Evaluate zz at each vertex: z(0,0)=0z(0,0) = 0, z(4,0)=12z(4,0) = 12, z(0,4)=8z(0,4) = 8.
  3. 3
    Step 3: Maximum is z=12z = 12 at (4,0)(4, 0).
  4. 4
    Check: The coefficient of xx is larger, so the maximum favors xx ✓

Answer

z=12z = 12 at (4,0)(4, 0)
Linear programming optimizes a linear objective over a feasible region defined by linear inequalities. The optimal value always occurs at a vertex (corner point) of the feasible region.

About Linear Programming

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

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More Linear Programming Examples

Example 2 hard

Minimize [formula] subject to [formula], [formula], [formula], [formula].

Example 3 easy

A feasible region has vertices at [formula], [formula], [formula], [formula]. Maximize [formula].

Example 4 medium

A company makes chairs ([formula]40[formula][formula] profit). Each chair takes 2 hours, each table

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