Question

A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function \( Z = 5x + 7y \), where \( x \) and \( y \) are the number of units of X and Y respectively sold. Which of the following statements is correct?

• A. The objective function maximizes the difference of the profit earned from products X and Y.
• B. The objective function measures the total production of products X and Y.
• C. The objective function maximizes the combined profit earned from selling X and Y.
• D. The objective function ensures the company produces more of product X than product Y.

Hint:

The objective function in linear programming typically represents a quantity to be maximized or minimized, such as profit, cost, or production.

Answer 1

The Correct Option is C

The given objective function is \( Z = 5x + 7y \). In this equation, \( Z \) represents the total profit earned by selling \( x \) units of product X and \( y \) units of product Y. Let's analyze the statements:

• The first option suggests the function maximizes the difference of the profit from X and Y. However, the expression \( Z = 5x + 7y \) is not a difference but a sum, indicating combined profits rather than differences.
• The second option implies the function measures total production of X and Y. This interpretation is incorrect because the coefficients 5 and 7 in the function correspond to profit contributions rather than quantities.
• The third option states the function maximizes the combined profit earned from X and Y. This aligns perfectly with the structure of the objective function \( Z = 5x + 7y \) since it sums total earnings, therefore maximizing profits from both products.
• The fourth option indicates that more of X than Y is produced, which is not addressed by the function since it lacks constraints or conditions that enforce the relation between \( x \) and \( y \).

Thus, the correct statement is: The objective function maximizes the combined profit earned from selling X and Y.