Question

Which of the following is the correct formulation of the linear programming problem ?

• A. Max \(Z=2x_1-x_2\) ;subject to \(x_1+x_2≤10;x_1≤3;x1≥0;x_2≤0 \)
• B. Max \(Z=3x_1+2x_2\) ;subject to \( x_1+2x2≥11;3x_1+x_2≥24;x_1≥0;x1,x_2≤0\)
• C. Max \(Z=2x_1+5x_2\) ;subject to \(4x_1+9x_2≤8;2x_1+3x_2≤9;x_1≥0;x_1,x_2≤0\)
• D. Min \(Z=4x_1+3x_2\) ;subject to \(x_1+9x_2≥8;2x_1+5x_2≤9;x_1≤0,x_2≥0 \)
• E. Min \(Z=x_1+5x_2\) ;subject to \(2x_1+5x_2≤10;x_1+3x_2≤9;x_1,x_2≥0\)

Answer 1

Answer: (E)

The correct formulation must satisfy these requirements:

1. All variables must have non-negativity constraints (x ≥ 0)
2. The constraints must form a convex feasible region
3. The objective function must be linear
4. Constraints must be properly balanced (≤ for maximization, ≥ for minimization typically)

Evaluating the options:

(A) Invalid because x₂ ≤ 0 violates non-negativity

(B) Invalid because x₁, x₂ ≤ 0 violates non-negativity

(E) Correct because:

• Minimization problem with proper constraints
• All variables satisfy x ≥ 0
• Constraints use ≤ appropriately

(D) Invalid because x₁ ≤ 0 violates non-negativity

(C) Invalid because x₁, x₂ < 0 violates non-negativity

The correct answer is (E) Min Z = x₁ + 5x₂; subject to 2x₁ + 5x₂ ≤ 10; x₁ + 3x₂ ≤ 9; x₁, x₂ ≥ 0.

Answer 2

A correct formulation of a linear programming problem requires:

1. A linear objective function: This function should be either maximized or minimized and be a linear combination of the decision variables.
2. Linear constraints: The constraints should be linear inequalities or equalities involving the decision variables.
3. Non-negativity restrictions: The decision variables must be non-negative (\(x_1, x_2, \dots \geq 0\)). While some problems allow for unrestricted variables, those can be handled by substitutions; the initial problem must not have variables with both positive and negative bounds simultaneously.

Examining each option we find that:

• Option 1: Has \(x_2 \leq 0\), meaning that \(x_2\) can take on negative values, and it is not stated that \(x_1\) must be positive. While feasible, it is not a standard LP form.
• Option 2: Has conflicting restrictions for \(x_1\) and \(x_2\). \(x_1\) must be non-negative, but also less than or equal to zero, which is a contradiction. The constraints are also inequalities in the wrong direction for a maximization problem (they should be less than or equal to).
• Option 3: Similar to Option 2; conflicting restrictions for \(x_1\) and \(x_2\).
• Option 4: Has conflicting restrictions for \(x_1\): \(x_1 \leq 0\) and \(x_1 \geq 0\) simultaneously.
• Option 5: This option satisfies all three conditions. It has a linear objective function to be minimized, linear constraints (all ≤), and non-negativity restrictions on the variables.

Therefore, only Option 5 is a correct formulation of a linear programming problem in standard form.

Final Answer: The final answer is \( \boxed{\text{Min } Z = x_1 + 5x_2; \text{ subject to } 2x_1 + 5x_2 \leq 10; \, x_1 + 3x_2 \leq 9; \, x_1, x_2 \geq 0} \).