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The primal of a LP problem is maximization of objective function with 6 variables and 2 constraints.
Which of the following correspond to the dual of the problem stated?
1. It has 2 variables and 6 constraints
2. It has 6 variables and 2 constraints
3. Maximization of objective function
4. Minimization of objective function
1. It has 2 variables and 6 constraints
2. It has 6 variables and 2 constraints
3. Maximization of objective function
4. Minimization of objective function
Select the correct answer using the codes given below:
- a)1 and 3
- b)1 and 4
- c)2 and 3
- d)2 and 4
Correct answer is option 'B'. Can you explain this answer?
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The primal of a LP problem is maximization of objective function with ...
Solution:
Given, primal of a LP problem is maximization of objective function with 6 variables and 2 constraints.
To find the dual of the problem, we can follow the below steps:
Step 1: Write the primal problem
Maximize Z = c₁x₁ + c₂x₂ + c₃x₃ + c₄x₄ + c₅x₅ + c₆x₆
Subject to,
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + a₁₄x₄ + a₁₅x₅ + a₁₆x₆ ≤ b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + a₂₄x₄ + a₂₅x₅ + a₂₆x₆ ≤ b₂
where x₁, x₂, x₃, x₄, x₅, x₆ are the decision variables, c₁, c₂, c₃, c₄, c₅, c₆ are the coefficients of the objective function, a₁₁, a₁₂, a₁₃, a₁₄, a₁₅, a₁₆, a₂₁, a₂₂, a₂₃, a₂₄, a₂₅, a₂₆ are the coefficients of the constraints, and b₁, b₂ are the right-hand side constants of the constraints.
Step 2: Write the dual problem
Minimize W = b₁y₁ + b₂y₂
Subject to,
a₁₁y₁ + a₂₁y₂ ≥ c₁
a₁₂y₁ + a₂₂y₂ ≥ c₂
a₁₃y₁ + a₂₃y₂ ≥ c₃
a₁₄y₁ + a₂₄y₂ ≥ c₄
a₁₅y₁ + a₂₅y₂ ≥ c₅
a₁₆y₁ + a₂₆y₂ ≥ c₆
where y₁, y₂ are the decision variables, b₁, b₂ are the coefficients of the objective function, c₁, c₂, c₃, c₄, c₅, c₆ are the coefficients of the constraints, and a₁₁, a₁₂, a₁₃, a₁₄, a₁₅, a₁₆, a₂₁, a₂₂, a₂₃, a₂₄, a₂₅, a₂₆ are the coefficients of the primal problem.
Step 3: Compare the dual problem with the given options
Option 1: It has 2 variables and 6 constraints
The dual problem has 2 variables and 6 constraints, which matches with this option.
Option 2: It has 6 variables and 2 constraints
The primal problem has 6 variables and 2 constraints, but the dual problem has 2 variables and 6 constraints, which does not match with this option.
Option 3: Maximization of objective function
The primal problem is a maximization problem, but the dual problem is a minimization problem, which does not match with this option.
Option 4: Minimization
Given, primal of a LP problem is maximization of objective function with 6 variables and 2 constraints.
To find the dual of the problem, we can follow the below steps:
Step 1: Write the primal problem
Maximize Z = c₁x₁ + c₂x₂ + c₃x₃ + c₄x₄ + c₅x₅ + c₆x₆
Subject to,
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + a₁₄x₄ + a₁₅x₅ + a₁₆x₆ ≤ b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + a₂₄x₄ + a₂₅x₅ + a₂₆x₆ ≤ b₂
where x₁, x₂, x₃, x₄, x₅, x₆ are the decision variables, c₁, c₂, c₃, c₄, c₅, c₆ are the coefficients of the objective function, a₁₁, a₁₂, a₁₃, a₁₄, a₁₅, a₁₆, a₂₁, a₂₂, a₂₃, a₂₄, a₂₅, a₂₆ are the coefficients of the constraints, and b₁, b₂ are the right-hand side constants of the constraints.
Step 2: Write the dual problem
Minimize W = b₁y₁ + b₂y₂
Subject to,
a₁₁y₁ + a₂₁y₂ ≥ c₁
a₁₂y₁ + a₂₂y₂ ≥ c₂
a₁₃y₁ + a₂₃y₂ ≥ c₃
a₁₄y₁ + a₂₄y₂ ≥ c₄
a₁₅y₁ + a₂₅y₂ ≥ c₅
a₁₆y₁ + a₂₆y₂ ≥ c₆
where y₁, y₂ are the decision variables, b₁, b₂ are the coefficients of the objective function, c₁, c₂, c₃, c₄, c₅, c₆ are the coefficients of the constraints, and a₁₁, a₁₂, a₁₃, a₁₄, a₁₅, a₁₆, a₂₁, a₂₂, a₂₃, a₂₄, a₂₅, a₂₆ are the coefficients of the primal problem.
Step 3: Compare the dual problem with the given options
Option 1: It has 2 variables and 6 constraints
The dual problem has 2 variables and 6 constraints, which matches with this option.
Option 2: It has 6 variables and 2 constraints
The primal problem has 6 variables and 2 constraints, but the dual problem has 2 variables and 6 constraints, which does not match with this option.
Option 3: Maximization of objective function
The primal problem is a maximization problem, but the dual problem is a minimization problem, which does not match with this option.
Option 4: Minimization
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The primal of a LP problem is maximization of objective function with 6 variables and 2 constraints.Which of the following correspond to the dual of the problem stated?1. It has 2 variables and 6 constraints2. It has 6 variables and 2 constraints3. Maximization of objective function4. Minimization of objective functionSelect the correct answer using the codes given below:a)1 and 3 b)1 and 4 c)2 and 3 d)2 and 4Correct answer is option 'B'. Can you explain this answer?
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