Consider the following problem.Maximize Z = 2x₁ + 4x₂ + 3x₃,subject to x₁ + 3x₂ + 2x₃ ≤ 30 x₁ + x₂ + x₃ ≤ 24 3x₁ + 5x₂ + 3x₃ ≤ 60 and x₁ ≥ 0,x₂ ≥ 0,x₃ ≥ 0.You are given the information that x₁ 0,x₂ = 0,and x₃ 0 in the optimal solution.Using the given information and the theory of the simplex method,analyze the constraints of the problem in order to identify a system of three constraint boundary equations (defining equations)whose simultaneous solution must be the optimal solution (not augmente d).Then solve this system of equations to obtain this solution.

Question

Consider the following problem.Maximize Z = 2x₁ + 4x₂ + 3x₃,subject to x₁ + 3x₂ + 2x₃ ≤ 30 x₁ + x₂ + x₃ ≤ 24 3x₁ + 5x₂ + 3x₃ ≤ 60 and x₁ ≥ 0,x₂ ≥ 0,x₃ ≥ 0.You are given the information that x₁ > 0,x₂ = 0,and x₃ >0 in the optimal solution.Using the given information and the theory of the simplex method,analyze the constraints of the problem in order to identify a system of three constraint boundary equations (defining equations)whose simultaneous solution must be the optimal solution (not augmente d).Then solve this system of equations to obtain this solution.

Quizplus · Accepted Answer

The Answer is Since x₁ > 0 and x₃ > 0,it follows that x₁ = 0 and x₃ = 0 cannot be part of the three defining equations at the optimal solution.Since x₂ = 0,then x₁ + x₃

\le

24 and 3 x₁ + 3 x₃

\le

60 or,equivalently,x₁ + x₃

\le

24 and x₁ + x₃

\le

20.This implies that the second constraint (x₁ + x₂ + x₃ ≤ 24)must have some slack,so its constraint boundary equation cannot be a defining equation at the optimal solution.We now have ruled out three of the six constraint boundary equations as being part of the defining equations at the optimal solution.For this three-variable problem,each CPF solution including the optimal solution must have three defining equations.Therefore,the three constraint boundary equations not ruled out must be the defining equations for the optimal solution.These defining equations are x₂ = 0,x₁ + 3 x₂ + 2 x₃ = 30,3 x₁ + 5 x₂ + 3 x₃ = 60.Solving this system of equations yields the optimal solution as (x₁,x₂,x₃)= (10,0,10).

Consider the Following Problem