Final answer:
Option a, Min 4x + 3y + (2/3)z, is the only properly stated valid objective function for a linear programming problem, as it is a linear combination of the variables with constant coefficients.
Step-by-step explanation:
The question is asking which of the provided options is a valid objective function for a linear programming problem. A valid objective function in linear programming must be a linear combination of the decision variables with constant coefficients. This means the function cannot include variables raised to any power other than one and cannot include any products of variables.
Let's evaluate the options:
- a. Min 4x + 3y + (2/3)z: This is a linear function since all the variables are to the first power and their coefficients are constant.
- b. Max 5x² + 6y²: This is not linear because variables are raised to the second power.
- c. Max 5xy: This is not a linear function because it involves a product of variables.
- d. Min (x₁+x₁)/3: This is a linear function (assuming it's a typo and meant to be (x + y)/3), simplified as (1/3)x + (1/3)y.
Therefore, options a and d are valid objective functions for a linear programming problem; however, given that option d has a typo, option a is the most suitable answer.